Relation homomorphisms, embeddings, isomorphisms

This file defines relation homomorphisms, embeddings, isomorphisms and order embeddings and isomorphisms.

Main declarations

• RelHom: Relation homomorphism. A RelHom r s is a function f : α → β such that r a b → s (f a) (f b).
• RelEmbedding: Relation embedding. A RelEmbedding r s is an embedding f : α ↪ β such that r a b ↔ s (f a) (f b).
• RelIso: Relation isomorphism. A RelIso r s is an equivalence f : α ≃ β such that r a b ↔ s (f a) (f b).
• sumLexCongr, prodLexCongr: Creates a relation homomorphism between two Sum.Lex or two Prod.Lex from relation homomorphisms between their arguments.

Notation

• →r: RelHom
• ↪r: RelEmbedding
• ≃r: RelIso

structure RelHom {α : Type u_5} {β : Type u_6} (r : α → α → Prop) (s : β → β → Prop) : Type (max u_5 u_6)

A relation homomorphism with respect to a given pair of relations r and s is a function f : α → β such that r a b → s (f a) (f b).

• toFun : α → β

  The underlying function of a RelHom

• map_rel' : ∀ {a b : α}, r a b → s (self.toFun a) (self.toFun b)

  A RelHom sends related elements to related elements

def «term_→r_» : Lean.TrailingParserDescr

A relation homomorphism with respect to a given pair of relations r and s is a function f : α → β such that r a b → s (f a) (f b).

Equations

• «term_→r_» = Lean.ParserDescr.trailingNode `term_→r_ 25 25 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " →r ") (Lean.ParserDescr.cat `term 26))

class RelHomClass (F : Type u_5) {α : Type u_6} {β : Type u_7} (r : outParam (α → α → Prop)) (s : outParam (β → β → Prop)) [FunLike F α β] : Prop

RelHomClass F r s asserts that F is a type of functions such that all f : F satisfy r a b → s (f a) (f b).

The relations r and s are outParams since figuring them out from a goal is a higher-order matching problem that Lean usually can't do unaided.

• map_rel : ∀ (f : F) {a b : α}, r a b → s (f a) (f b)

  A RelHomClass sends related elements to related elements

theorem RelHomClass.isIrrefl {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {F : Type u_5} [FunLike F α β] [RelHomClass F r s] (f : F) [IsIrrefl β s] : IsIrrefl α r

theorem RelHomClass.isAsymm {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {F : Type u_5} [FunLike F α β] [RelHomClass F r s] (f : F) [IsAsymm β s] : IsAsymm α r

theorem RelHomClass.acc {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {F : Type u_5} [FunLike F α β] [RelHomClass F r s] (f : F) (a : α) : Acc s (f a) → Acc r a

theorem RelHomClass.wellFounded {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {F : Type u_5} [FunLike F α β] [RelHomClass F r s] (f : F) : WellFounded s → WellFounded r

instance RelHom.instFunLikeRelHom {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : FunLike (r →r s) α β

Equations

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

instance RelHom.instRelHomClassRelHomInstFunLikeRelHom {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : RelHomClass (r →r s) r s

Equations

• (_ : RelHomClass (r →r s) r s) = (_ : RelHomClass (r →r s) r s)

theorem RelHom.map_rel {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r →r s) {a : α} {b : α} : r a b → s (f a) (f b)

@[simp]

theorem RelHom.coe_fn_toFun {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r →r s) : f.toFun = ⇑f

theorem RelHom.coe_fn_injective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : Function.Injective fun (f : r →r s) => ⇑f

The map coe_fn : (r →r s) → (α → β) is injective.

theorem RelHom.ext {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} ⦃f : r →r s⦄ ⦃g : r →r s⦄ (h : ∀ (x : α), f x = g x) : f = g

theorem RelHom.ext_iff {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {f : r →r s} {g : r →r s} : f = g ↔ ∀ (x : α), f x = g x

@[simp]

theorem RelHom.id_apply {α : Type u_1} (r : α → α → Prop) (x : α) : (RelHom.id r) x = x

def RelHom.id {α : Type u_1} (r : α → α → Prop) : r →r r

Identity map is a relation homomorphism.

Equations

• RelHom.id r = { toFun := fun (x : α) => x, map_rel' := (_ : ∀ {a b : α}, r a b → r a b) }

@[simp]

theorem RelHom.comp_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (g : s →r t) (f : r →r s) (x : α) : (RelHom.comp g f) x = g (f x)

def RelHom.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (g : s →r t) (f : r →r s) : r →r t

Composition of two relation homomorphisms is a relation homomorphism.

Equations

• RelHom.comp g f = { toFun := fun (x : α) => g (f x), map_rel' := (_ : ∀ {a b : α}, r a b → t (g.toFun (f a)) (g.toFun (f b))) }

def RelHom.swap {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r →r s) : Function.swap r →r Function.swap s

A relation homomorphism is also a relation homomorphism between dual relations.

Equations

• RelHom.swap f = { toFun := ⇑f, map_rel' := (_ : ∀ {a b : α}, r b a → s (f b) (f a)) }

def RelHom.preimage {α : Type u_1} {β : Type u_2} (f : α → β) (s : β → β → Prop) : f ⁻¹'o s →r s

A function is a relation homomorphism from the preimage relation of s to s.

Equations

• RelHom.preimage f s = { toFun := f, map_rel' := (_ : ∀ {a b : α}, (f ⁻¹'o s) a b → (f ⁻¹'o s) a b) }

theorem injective_of_increasing {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsTrichotomous α r] [IsIrrefl β s] (f : α → β) (hf : ∀ {x y : α}, r x y → s (f x) (f y)) : Function.Injective f

An increasing function is injective

theorem RelHom.injective_of_increasing {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsTrichotomous α r] [IsIrrefl β s] (f : r →r s) : Function.Injective ⇑f

An increasing function is injective

theorem Function.Surjective.wellFounded_iff {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {f : α → β} (hf : Function.Surjective f) (o : ∀ {a b : α}, r a b ↔ s (f a) (f b)) : WellFounded r ↔ WellFounded s

structure RelEmbedding {α : Type u_5} {β : Type u_6} (r : α → α → Prop) (s : β → β → Prop) extends Function.Embedding : Type (max u_5 u_6)

A relation embedding with respect to a given pair of relations r and s is an embedding f : α ↪ β such that r a b ↔ s (f a) (f b).

• toFun : α → β
• inj' : Function.Injective self.toFun
• map_rel_iff' : ∀ {a b : α}, s (self.toEmbedding a) (self.toEmbedding b) ↔ r a b

  Elements are related iff they are related after apply a RelEmbedding

def «term_↪r_» : Lean.TrailingParserDescr

A relation embedding with respect to a given pair of relations r and s is an embedding f : α ↪ β such that r a b ↔ s (f a) (f b).

Equations

• «term_↪r_» = Lean.ParserDescr.trailingNode `term_↪r_ 25 25 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ↪r ") (Lean.ParserDescr.cat `term 26))

def Subtype.relEmbedding {X : Type u_5} (r : X → X → Prop) (p : X → Prop) : Subtype.val ⁻¹'o r ↪r r

The induced relation on a subtype is an embedding under the natural inclusion.

Equations

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

theorem preimage_equivalence {α : Sort u_5} {β : Sort u_6} (f : α → β) {s : β → β → Prop} (hs : Equivalence s) : Equivalence (f ⁻¹'o s)

def RelEmbedding.toRelHom {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) : r →r s

A relation embedding is also a relation homomorphism

Equations

• RelEmbedding.toRelHom f = { toFun := f.toFun, map_rel' := (_ : ∀ {a b : α}, r a b → s (f.toEmbedding a) (f.toEmbedding b)) }

instance RelEmbedding.instCoeRelEmbeddingRelHom {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : Coe (r ↪r s) (r →r s)

Equations

• RelEmbedding.instCoeRelEmbeddingRelHom = { coe := RelEmbedding.toRelHom }

instance RelEmbedding.instFunLikeRelEmbedding {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : FunLike (r ↪r s) α β

Equations

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

instance RelEmbedding.instRelHomClassRelEmbeddingInstFunLikeRelEmbedding {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : RelHomClass (r ↪r s) r s

Equations

• (_ : RelHomClass (r ↪r s) r s) = (_ : RelHomClass (r ↪r s) r s)

instance RelEmbedding.instEmbeddingLikeRelEmbeddingInstFunLikeRelEmbedding {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : EmbeddingLike (r ↪r s) α β

Equations

• (_ : EmbeddingLike (r ↪r s) α β) = (_ : EmbeddingLike (r ↪r s) α β)

@[simp]

theorem RelEmbedding.coe_toEmbedding {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {f : r ↪r s} : ⇑f.toEmbedding = ⇑f

@[simp]

theorem RelEmbedding.coe_toRelHom {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {f : r ↪r s} : ⇑(RelEmbedding.toRelHom f) = ⇑f

theorem RelEmbedding.injective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) : Function.Injective ⇑f

theorem RelEmbedding.inj {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) {a : α} {b : α} : f a = f b ↔ a = b

theorem RelEmbedding.map_rel_iff {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) {a : α} {b : α} : s (f a) (f b) ↔ r a b

@[simp]

theorem RelEmbedding.coe_mk {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {f : α ↪ β} {h : ∀ {a b : α}, s (f a) (f b) ↔ r a b} : ⇑{ toEmbedding := f, map_rel_iff' := h } = ⇑f

theorem RelEmbedding.coe_fn_injective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : Function.Injective fun (f : r ↪r s) => ⇑f

The map coe_fn : (r ↪r s) → (α → β) is injective.

theorem RelEmbedding.ext {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} ⦃f : r ↪r s⦄ ⦃g : r ↪r s⦄ (h : ∀ (x : α), f x = g x) : f = g

theorem RelEmbedding.ext_iff {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {f : r ↪r s} {g : r ↪r s} : f = g ↔ ∀ (x : α), f x = g x

@[simp]

theorem RelEmbedding.refl_apply {α : Type u_1} (r : α → α → Prop) (a : α) : (RelEmbedding.refl r) a = a

def RelEmbedding.refl {α : Type u_1} (r : α → α → Prop) : r ↪r r

Identity map is a relation embedding.

Equations

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

def RelEmbedding.trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (f : r ↪r s) (g : s ↪r t) : r ↪r t

Composition of two relation embeddings is a relation embedding.

Equations

• RelEmbedding.trans f g = { toEmbedding := Function.Embedding.trans f.toEmbedding g.toEmbedding, map_rel_iff' := (_ : ∀ (a a_1 : α), t (g (f a)) (g (f a_1)) ↔ r a a_1) }

instance RelEmbedding.instInhabitedRelEmbedding {α : Type u_1} (r : α → α → Prop) : Inhabited (r ↪r r)

Equations

• RelEmbedding.instInhabitedRelEmbedding r = { default := RelEmbedding.refl r }

theorem RelEmbedding.trans_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (f : r ↪r s) (g : s ↪r t) (a : α) : (RelEmbedding.trans f g) a = g (f a)

@[simp]

theorem RelEmbedding.coe_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (f : r ↪r s) (g : s ↪r t) : ⇑(RelEmbedding.trans f g) = ⇑g ∘ ⇑f

def RelEmbedding.swap {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) : Function.swap r ↪r Function.swap s

A relation embedding is also a relation embedding between dual relations.

Equations

• RelEmbedding.swap f = { toEmbedding := f.toEmbedding, map_rel_iff' := (_ : ∀ {a b : α}, s (f b) (f a) ↔ r b a) }

def RelEmbedding.preimage {α : Type u_1} {β : Type u_2} (f : α ↪ β) (s : β → β → Prop) : ⇑f ⁻¹'o s ↪r s

If f is injective, then it is a relation embedding from the preimage relation of s to s.

Equations

• RelEmbedding.preimage f s = { toEmbedding := f, map_rel_iff' := (_ : ∀ {a b : α}, s (f a) (f b) ↔ s (f a) (f b)) }

theorem RelEmbedding.eq_preimage {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) : r = ⇑f ⁻¹'o s

theorem RelEmbedding.isIrrefl {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) [IsIrrefl β s] : IsIrrefl α r

theorem RelEmbedding.isRefl {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) [IsRefl β s] : IsRefl α r

theorem RelEmbedding.isSymm {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) [IsSymm β s] : IsSymm α r

theorem RelEmbedding.isAsymm {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) [IsAsymm β s] : IsAsymm α r

theorem RelEmbedding.isAntisymm {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : r ↪r s → ∀ [inst : IsAntisymm β s], IsAntisymm α r

theorem RelEmbedding.isTrans {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : r ↪r s → ∀ [inst : IsTrans β s], IsTrans α r

theorem RelEmbedding.isTotal {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : r ↪r s → ∀ [inst : IsTotal β s], IsTotal α r

theorem RelEmbedding.isPreorder {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : r ↪r s → ∀ [inst : IsPreorder β s], IsPreorder α r

theorem RelEmbedding.isPartialOrder {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : r ↪r s → ∀ [inst : IsPartialOrder β s], IsPartialOrder α r

theorem RelEmbedding.isLinearOrder {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : r ↪r s → ∀ [inst : IsLinearOrder β s], IsLinearOrder α r

theorem RelEmbedding.isStrictOrder {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : r ↪r s → ∀ [inst : IsStrictOrder β s], IsStrictOrder α r

theorem RelEmbedding.isTrichotomous {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : r ↪r s → ∀ [inst : IsTrichotomous β s], IsTrichotomous α r

theorem RelEmbedding.isStrictTotalOrder {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : r ↪r s → ∀ [inst : IsStrictTotalOrder β s], IsStrictTotalOrder α r

theorem RelEmbedding.acc {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) (a : α) : Acc s (f a) → Acc r a

theorem RelEmbedding.wellFounded {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : r ↪r s → WellFounded s → WellFounded r

theorem RelEmbedding.isWellFounded {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) [IsWellFounded β s] : IsWellFounded α r

theorem RelEmbedding.isWellOrder {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : r ↪r s → ∀ [inst : IsWellOrder β s], IsWellOrder α r

instance Subtype.wellFoundedLT {α : Type u_1} [LT α] [WellFoundedLT α] (p : α → Prop) : WellFoundedLT (Subtype p)

Equations

• (_ : WellFoundedLT (Subtype p)) = (_ : IsWellFounded (Subtype p) (Subtype.val ⁻¹'o fun (x x_1 : α) => x < x_1))

instance Subtype.wellFoundedGT {α : Type u_1} [LT α] [WellFoundedGT α] (p : α → Prop) : WellFoundedGT (Subtype p)

Equations

• (_ : WellFoundedGT (Subtype p)) = (_ : IsWellFounded (Subtype p) (Subtype.val ⁻¹'o fun (x x_1 : α) => x > x_1))

@[simp]

theorem Quotient.mkRelHom_apply {α : Type u_1} [Setoid α] {r : α → α → Prop} (H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂) (a : α) : (Quotient.mkRelHom H) a = Quotient.mk' a

def Quotient.mkRelHom {α : Type u_1} [Setoid α] {r : α → α → Prop} (H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂) : r →r Quotient.lift₂ r H

Quotient.mk' as a relation homomorphism between the relation and the lift of a relation.

Equations

• Quotient.mkRelHom H = { toFun := Quotient.mk', map_rel' := (_ : ∀ {a b : α}, r a b → r a b) }

@[simp]

theorem Quotient.outRelEmbedding_apply {α : Type u_1} [Setoid α] {r : α → α → Prop} (H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂) : ∀ (a : Quotient inst✝), (Quotient.outRelEmbedding H) a = Quotient.out a

noncomputable def Quotient.outRelEmbedding {α : Type u_1} [Setoid α] {r : α → α → Prop} (H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂) : Quotient.lift₂ r H ↪r r

Quotient.out as a relation embedding between the lift of a relation and the relation.

Equations

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

@[simp]

theorem Quotient.out'RelEmbedding_apply {α : Type u_1} : ∀ {x : Setoid α} {r : α → α → Prop} (H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂) (a : Quotient x), (Quotient.out'RelEmbedding H) a = Quotient.out' a

noncomputable def Quotient.out'RelEmbedding {α : Type u_1} : {x : Setoid α} → {r : α → α → Prop} → (H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂) → (fun (a b : Quotient x) => Quotient.liftOn₂' a b r H) ↪r r

Quotient.out' as a relation embedding between the lift of a relation and the relation.

Equations

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

@[simp]

theorem acc_lift₂_iff {α : Type u_1} [Setoid α] {r : α → α → Prop} {H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂} {a : α} : Acc (Quotient.lift₂ r H) ⟦a⟧ ↔ Acc r a

@[simp]

theorem acc_liftOn₂'_iff {α : Type u_1} {s : Setoid α} {r : α → α → Prop} {H : ∀ (a₁ a₂ b₁ b₂ : α), Setoid.r a₁ b₁ → Setoid.r a₂ b₂ → r a₁ a₂ = r b₁ b₂} {a : α} : Acc (fun (x y : Quotient s) => Quotient.liftOn₂' x y r H) (Quotient.mk'' a) ↔ Acc r a

@[simp]

theorem wellFounded_lift₂_iff {α : Type u_1} [Setoid α] {r : α → α → Prop} {H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂} : WellFounded (Quotient.lift₂ r H) ↔ WellFounded r

A relation is well founded iff its lift to a quotient is.

theorem WellFounded.quotient_lift₂ {α : Type u_1} [Setoid α] {r : α → α → Prop} {H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂} : WellFounded r → WellFounded (Quotient.lift₂ r H)

Alias of the reverse direction of wellFounded_lift₂_iff.

---

A relation is well founded iff its lift to a quotient is.

theorem WellFounded.of_quotient_lift₂ {α : Type u_1} [Setoid α] {r : α → α → Prop} {H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂} : WellFounded (Quotient.lift₂ r H) → WellFounded r

Alias of the forward direction of wellFounded_lift₂_iff.

---

A relation is well founded iff its lift to a quotient is.

@[simp]

theorem wellFounded_liftOn₂'_iff {α : Type u_1} {s : Setoid α} {r : α → α → Prop} {H : ∀ (a₁ a₂ b₁ b₂ : α), Setoid.r a₁ b₁ → Setoid.r a₂ b₂ → r a₁ a₂ = r b₁ b₂} : (WellFounded fun (x y : Quotient s) => Quotient.liftOn₂' x y r H) ↔ WellFounded r

theorem WellFounded.of_quotient_liftOn₂' {α : Type u_1} {s : Setoid α} {r : α → α → Prop} {H : ∀ (a₁ a₂ b₁ b₂ : α), Setoid.r a₁ b₁ → Setoid.r a₂ b₂ → r a₁ a₂ = r b₁ b₂} : (WellFounded fun (x y : Quotient s) => Quotient.liftOn₂' x y r H) → WellFounded r

Alias of the forward direction of wellFounded_liftOn₂'_iff.

theorem WellFounded.quotient_liftOn₂' {α : Type u_1} {s : Setoid α} {r : α → α → Prop} {H : ∀ (a₁ a₂ b₁ b₂ : α), Setoid.r a₁ b₁ → Setoid.r a₂ b₂ → r a₁ a₂ = r b₁ b₂} : WellFounded r → WellFounded fun (x y : Quotient s) => Quotient.liftOn₂' x y r H

Alias of the reverse direction of wellFounded_liftOn₂'_iff.

def RelEmbedding.ofMapRelIff {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : α → β) [IsAntisymm α r] [IsRefl β s] (hf : ∀ (a b : α), s (f a) (f b) ↔ r a b) : r ↪r s

To define a relation embedding from an antisymmetric relation r to a reflexive relation s it suffices to give a function together with a proof that it satisfies s (f a) (f b) ↔ r a b.

Equations

• RelEmbedding.ofMapRelIff f hf = { toEmbedding := { toFun := f, inj' := (_ : ∀ (x x_1 : α), f x = f x_1 → x = x_1) }, map_rel_iff' := (_ : ∀ {a b : α}, s (f a) (f b) ↔ r a b) }

@[simp]

theorem RelEmbedding.ofMapRelIff_coe {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : α → β) [IsAntisymm α r] [IsRefl β s] (hf : ∀ (a b : α), s (f a) (f b) ↔ r a b) : ⇑(RelEmbedding.ofMapRelIff f hf) = f

def RelEmbedding.ofMonotone {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsTrichotomous α r] [IsAsymm β s] (f : α → β) (H : ∀ (a b : α), r a b → s (f a) (f b)) : r ↪r s

It suffices to prove f is monotone between strict relations to show it is a relation embedding.

Equations

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

@[simp]

theorem RelEmbedding.ofMonotone_coe {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsTrichotomous α r] [IsAsymm β s] (f : α → β) (H : ∀ (a b : α), r a b → s (f a) (f b)) : ⇑(RelEmbedding.ofMonotone f H) = f

def RelEmbedding.ofIsEmpty {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsEmpty α] : r ↪r s

A relation embedding from an empty type.

Equations

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

@[simp]

theorem RelEmbedding.sumLiftRelInl_apply {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) (val : α) : (RelEmbedding.sumLiftRelInl r s) val = Sum.inl val

def RelEmbedding.sumLiftRelInl {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) : r ↪r Sum.LiftRel r s

Sum.inl as a relation embedding into Sum.LiftRel r s.

Equations

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

@[simp]

theorem RelEmbedding.sumLiftRelInr_apply {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) (val : β) : (RelEmbedding.sumLiftRelInr r s) val = Sum.inr val

def RelEmbedding.sumLiftRelInr {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) : s ↪r Sum.LiftRel r s

Sum.inr as a relation embedding into Sum.LiftRel r s.

Equations

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

@[simp]

theorem RelEmbedding.sumLiftRelMap_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} {u : δ → δ → Prop} (f : r ↪r s) (g : t ↪r u) : ∀ (a : α ⊕ γ), (RelEmbedding.sumLiftRelMap f g) a = Sum.map (⇑f) (⇑g) a

def RelEmbedding.sumLiftRelMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} {u : δ → δ → Prop} (f : r ↪r s) (g : t ↪r u) : Sum.LiftRel r t ↪r Sum.LiftRel s u

Sum.map as a relation embedding between Sum.LiftRel relations.

Equations

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

@[simp]

theorem RelEmbedding.sumLexInl_apply {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) (val : α) : (RelEmbedding.sumLexInl r s) val = Sum.inl val

def RelEmbedding.sumLexInl {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) : r ↪r Sum.Lex r s

Sum.inl as a relation embedding into Sum.Lex r s.

Equations

• RelEmbedding.sumLexInl r s = { toEmbedding := { toFun := Sum.inl, inj' := (_ : Function.Injective Sum.inl) }, map_rel_iff' := (_ : ∀ {a b : α}, Sum.Lex r s (Sum.inl a) (Sum.inl b) ↔ r a b) }

@[simp]

theorem RelEmbedding.sumLexInr_apply {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) (val : β) : (RelEmbedding.sumLexInr r s) val = Sum.inr val

def RelEmbedding.sumLexInr {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) : s ↪r Sum.Lex r s

Sum.inr as a relation embedding into Sum.Lex r s.

Equations

• RelEmbedding.sumLexInr r s = { toEmbedding := { toFun := Sum.inr, inj' := (_ : Function.Injective Sum.inr) }, map_rel_iff' := (_ : ∀ {a b : β}, Sum.Lex r s (Sum.inr a) (Sum.inr b) ↔ s a b) }

@[simp]

theorem RelEmbedding.sumLexMap_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} {u : δ → δ → Prop} (f : r ↪r s) (g : t ↪r u) : ∀ (a : α ⊕ γ), (RelEmbedding.sumLexMap f g) a = Sum.map (⇑f) (⇑g) a

def RelEmbedding.sumLexMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} {u : δ → δ → Prop} (f : r ↪r s) (g : t ↪r u) : Sum.Lex r t ↪r Sum.Lex s u

Sum.map as a relation embedding between Sum.Lex relations.

Equations

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

@[simp]

theorem RelEmbedding.prodLexMkLeft_apply {α : Type u_1} {β : Type u_2} {r : α → α → Prop} (s : β → β → Prop) {a : α} (h : ¬r a a) (snd : β) : (RelEmbedding.prodLexMkLeft s h) snd = (a, snd)

def RelEmbedding.prodLexMkLeft {α : Type u_1} {β : Type u_2} {r : α → α → Prop} (s : β → β → Prop) {a : α} (h : ¬r a a) : s ↪r Prod.Lex r s

fun b ↦ Prod.mk a b as a relation embedding.

Equations

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

@[simp]

theorem RelEmbedding.prodLexMkRight_apply {α : Type u_1} {β : Type u_2} {s : β → β → Prop} (r : α → α → Prop) {b : β} (h : ¬s b b) (a : α) : (RelEmbedding.prodLexMkRight r h) a = (a, b)

def RelEmbedding.prodLexMkRight {α : Type u_1} {β : Type u_2} {s : β → β → Prop} (r : α → α → Prop) {b : β} (h : ¬s b b) : r ↪r Prod.Lex r s

fun a ↦ Prod.mk a b as a relation embedding.

Equations

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

@[simp]

theorem RelEmbedding.prodLexMap_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} {u : δ → δ → Prop} (f : r ↪r s) (g : t ↪r u) : ∀ (a : α × γ), (RelEmbedding.prodLexMap f g) a = Prod.map (⇑f) (⇑g) a

def RelEmbedding.prodLexMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} {u : δ → δ → Prop} (f : r ↪r s) (g : t ↪r u) : Prod.Lex r t ↪r Prod.Lex s u

Prod.map as a relation embedding.

Equations

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

structure RelIso {α : Type u_5} {β : Type u_6} (r : α → α → Prop) (s : β → β → Prop) extends Equiv : Type (max u_5 u_6)

A relation isomorphism is an equivalence that is also a relation embedding.

• toFun : α → β
• invFun : β → α
• left_inv : Function.LeftInverse self.invFun self.toFun
• right_inv : Function.RightInverse self.invFun self.toFun
• map_rel_iff' : ∀ {a b : α}, s (self.toEquiv a) (self.toEquiv b) ↔ r a b

  Elements are related iff they are related after apply a RelIso

def «term_≃r_» : Lean.TrailingParserDescr

A relation isomorphism is an equivalence that is also a relation embedding.

Equations

• «term_≃r_» = Lean.ParserDescr.trailingNode `term_≃r_ 25 25 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ≃r ") (Lean.ParserDescr.cat `term 26))

def RelIso.toRelEmbedding {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) : r ↪r s

Convert a RelIso to a RelEmbedding. This function is also available as a coercion but often it is easier to write f.toRelEmbedding than to write explicitly r and s in the target type.

Equations

• RelIso.toRelEmbedding f = { toEmbedding := Equiv.toEmbedding f.toEquiv, map_rel_iff' := (_ : ∀ {a b : α}, s (f.toEquiv a) (f.toEquiv b) ↔ r a b) }

theorem RelIso.toEquiv_injective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : Function.Injective RelIso.toEquiv

instance RelIso.instCoeOutRelIsoRelEmbedding {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : CoeOut (r ≃r s) (r ↪r s)

Equations

• RelIso.instCoeOutRelIsoRelEmbedding = { coe := RelIso.toRelEmbedding }

instance RelIso.instFunLikeRelIso {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : FunLike (r ≃r s) α β

Equations

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

instance RelIso.instRelHomClassRelIsoInstFunLikeRelIso {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : RelHomClass (r ≃r s) r s

Equations

• (_ : RelHomClass (r ≃r s) r s) = (_ : RelHomClass (r ≃r s) r s)

instance RelIso.instEquivLikeRelIso {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : EquivLike (r ≃r s) α β

Equations

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

instance RelIso.instCoeFunRelIsoForAll {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : CoeFun (r ≃r s) fun (x : r ≃r s) => α → β

Equations

• RelIso.instCoeFunRelIsoForAll = { coe := DFunLike.coe }

@[simp]

theorem RelIso.coe_toRelEmbedding {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) : ⇑(RelIso.toRelEmbedding f) = ⇑f

@[simp]

theorem RelIso.coe_toEmbedding {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) : ⇑(Equiv.toEmbedding f.toEquiv) = ⇑f

theorem RelIso.map_rel_iff {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) {a : α} {b : α} : s (f a) (f b) ↔ r a b

@[simp]

theorem RelIso.coe_fn_mk {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : α ≃ β) (o : ∀ ⦃a b : α⦄, s (f a) (f b) ↔ r a b) : ⇑{ toEquiv := f, map_rel_iff' := o } = ⇑f

@[simp]

theorem RelIso.coe_fn_toEquiv {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) : ⇑f.toEquiv = ⇑f

theorem RelIso.coe_fn_injective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} : Function.Injective fun (f : r ≃r s) => ⇑f

The map coe_fn : (r ≃r s) → (α → β) is injective. Lean fails to parse function.injective (λ e : r ≃r s, (e : α → β)), so we use a trick to say the same.

theorem RelIso.ext {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} ⦃f : r ≃r s⦄ ⦃g : r ≃r s⦄ (h : ∀ (x : α), f x = g x) : f = g

theorem RelIso.ext_iff {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {f : r ≃r s} {g : r ≃r s} : f = g ↔ ∀ (x : α), f x = g x

def RelIso.symm {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) : s ≃r r

Inverse map of a relation isomorphism is a relation isomorphism.

Equations

• RelIso.symm f = { toEquiv := f.symm, map_rel_iff' := (_ : ∀ (a b : β), r (f.symm a) (f.symm b) ↔ s a b) }

def RelIso.Simps.apply {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (h : r ≃r s) : α → β

See Note [custom simps projection]. We need to specify this projection explicitly in this case, because RelIso defines custom coercions other than the ones given by DFunLike.

Equations

• RelIso.Simps.apply h = ⇑h

def RelIso.Simps.symm_apply {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (h : r ≃r s) : β → α

See Note [custom simps projection].

Equations

• RelIso.Simps.symm_apply h = ⇑(RelIso.symm h)

@[simp]

theorem RelIso.refl_apply {α : Type u_1} (r : α → α → Prop) (a : α) : (RelIso.refl r) a = a

def RelIso.refl {α : Type u_1} (r : α → α → Prop) : r ≃r r

Identity map is a relation isomorphism.

Equations

• RelIso.refl r = { toEquiv := Equiv.refl α, map_rel_iff' := (_ : ∀ {a b : α}, r ((Equiv.refl α) a) ((Equiv.refl α) b) ↔ r ((Equiv.refl α) a) ((Equiv.refl α) b)) }

@[simp]

theorem RelIso.trans_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (f₁ : r ≃r s) (f₂ : s ≃r t) : ∀ (a : α), (RelIso.trans f₁ f₂) a = f₂ (f₁ a)

def RelIso.trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (f₁ : r ≃r s) (f₂ : s ≃r t) : r ≃r t

Composition of two relation isomorphisms is a relation isomorphism.

Equations

• RelIso.trans f₁ f₂ = { toEquiv := f₁.trans f₂.toEquiv, map_rel_iff' := (_ : ∀ {a b : α}, t (f₂ (f₁.toEquiv a)) (f₂ (f₁.toEquiv b)) ↔ r a b) }

instance RelIso.instInhabitedRelIso {α : Type u_1} (r : α → α → Prop) : Inhabited (r ≃r r)

Equations

• RelIso.instInhabitedRelIso r = { default := RelIso.refl r }

@[simp]

theorem RelIso.default_def {α : Type u_1} (r : α → α → Prop) : default = RelIso.refl r

@[simp]

theorem RelIso.cast_toEquiv {α : Type u} {β : Type u} {r : α → α → Prop} {s : β → β → Prop} (h₁ : α = β) (h₂ : HEq r s) : (RelIso.cast h₁ h₂).toEquiv = Equiv.cast h₁

@[simp]

theorem RelIso.cast_apply {α : Type u} {β : Type u} {r : α → α → Prop} {s : β → β → Prop} (h₁ : α = β) (h₂ : HEq r s) (a : α) : (RelIso.cast h₁ h₂) a = cast h₁ a

def RelIso.cast {α : Type u} {β : Type u} {r : α → α → Prop} {s : β → β → Prop} (h₁ : α = β) (h₂ : HEq r s) : r ≃r s

A relation isomorphism between equal relations on equal types.

Equations

• RelIso.cast h₁ h₂ = { toEquiv := Equiv.cast h₁, map_rel_iff' := (_ : ∀ (a b : α), s ((Equiv.cast h₁) a) ((Equiv.cast h₁) b) ↔ r a b) }

@[simp]

theorem RelIso.cast_symm {α : Type u} {β : Type u} {r : α → α → Prop} {s : β → β → Prop} (h₁ : α = β) (h₂ : HEq r s) : RelIso.symm (RelIso.cast h₁ h₂) = RelIso.cast (_ : β = α) (_ : HEq s r)

@[simp]

theorem RelIso.cast_refl {α : Type u} {r : α → α → Prop} (h₁ : optParam (α = α) (_ : α = α)) (h₂ : optParam (HEq r r) (_ : HEq r r)) : RelIso.cast h₁ h₂ = RelIso.refl r

@[simp]

theorem RelIso.cast_trans {α : Type u} {β : Type u} {γ : Type u} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (h₁ : α = β) (h₁' : β = γ) (h₂ : HEq r s) (h₂' : HEq s t) : RelIso.trans (RelIso.cast h₁ h₂) (RelIso.cast h₁' h₂') = RelIso.cast (_ : α = γ) (_ : HEq r t)

def RelIso.swap {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) : Function.swap r ≃r Function.swap s

a relation isomorphism is also a relation isomorphism between dual relations.

Equations

• RelIso.swap f = { toEquiv := f.toEquiv, map_rel_iff' := (_ : ∀ {a b : α}, s (f b) (f a) ↔ r b a) }

@[simp]

theorem RelIso.coe_fn_symm_mk {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : α ≃ β) (o : ∀ {a b : α}, s (f a) (f b) ↔ r a b) : ⇑(RelIso.symm { toEquiv := f, map_rel_iff' := o }) = ⇑f.symm

@[simp]

theorem RelIso.apply_symm_apply {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) (x : β) : e ((RelIso.symm e) x) = x

@[simp]

theorem RelIso.symm_apply_apply {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) (x : α) : (RelIso.symm e) (e x) = x

theorem RelIso.rel_symm_apply {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) {x : α} {y : β} : r x ((RelIso.symm e) y) ↔ s (e x) y

theorem RelIso.symm_apply_rel {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) {x : β} {y : α} : r ((RelIso.symm e) x) y ↔ s x (e y)

theorem RelIso.bijective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) : Function.Bijective ⇑e

theorem RelIso.injective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) : Function.Injective ⇑e

theorem RelIso.surjective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) : Function.Surjective ⇑e

theorem RelIso.eq_iff_eq {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) {a : α} {b : α} : f a = f b ↔ a = b

def RelIso.preimage {α : Type u_1} {β : Type u_2} (f : α ≃ β) (s : β → β → Prop) : ⇑f ⁻¹'o s ≃r s

Any equivalence lifts to a relation isomorphism between s and its preimage.

Equations

• RelIso.preimage f s = { toEquiv := f, map_rel_iff' := (_ : ∀ {a b : α}, s (f a) (f b) ↔ s (f a) (f b)) }

instance RelIso.IsWellOrder.preimage {β : Type u_2} {α : Type u} (r : α → α → Prop) [IsWellOrder α r] (f : β ≃ α) : IsWellOrder β (⇑f ⁻¹'o r)

Equations

• (_ : IsWellOrder β (⇑f ⁻¹'o r)) = (_ : IsWellOrder β (⇑f ⁻¹'o r))

instance RelIso.IsWellOrder.ulift {α : Type u} (r : α → α → Prop) [IsWellOrder α r] : IsWellOrder (ULift.{u_5, u} α) (ULift.down ⁻¹'o r)

Equations

• (_ : IsWellOrder (ULift.{u_5, u} α) (ULift.down ⁻¹'o r)) = (_ : IsWellOrder (ULift.{u_5, u} α) (⇑Equiv.ulift ⁻¹'o r))

@[simp]

theorem RelIso.ofSurjective_apply {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) (H : Function.Surjective ⇑f) (a : α) : (RelIso.ofSurjective f H) a = f a

noncomputable def RelIso.ofSurjective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) (H : Function.Surjective ⇑f) : r ≃r s

A surjective relation embedding is a relation isomorphism.

Equations

• RelIso.ofSurjective f H = { toEquiv := Equiv.ofBijective ⇑f (_ : Function.Injective ⇑f ∧ Function.Surjective ⇑f), map_rel_iff' := (_ : ∀ {a b : α}, s (f a) (f b) ↔ r a b) }

def RelIso.sumLexCongr {α₁ : Type u_5} {α₂ : Type u_6} {β₁ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₁ : β₁ → β₁ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r s₁) (e₂ : r₂ ≃r s₂) : Sum.Lex r₁ r₂ ≃r Sum.Lex s₁ s₂

Given relation isomorphisms r₁ ≃r s₁ and r₂ ≃r s₂, construct a relation isomorphism for the lexicographic orders on the sum.

Equations

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

def RelIso.prodLexCongr {α₁ : Type u_5} {α₂ : Type u_6} {β₁ : Type u_7} {β₂ : Type u_8} {r₁ : α₁ → α₁ → Prop} {r₂ : α₂ → α₂ → Prop} {s₁ : β₁ → β₁ → Prop} {s₂ : β₂ → β₂ → Prop} (e₁ : r₁ ≃r s₁) (e₂ : r₂ ≃r s₂) : Prod.Lex r₁ r₂ ≃r Prod.Lex s₁ s₂

Given relation isomorphisms r₁ ≃r s₁ and r₂ ≃r s₂, construct a relation isomorphism for the lexicographic orders on the product.

Equations

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

def RelIso.relIsoOfIsEmpty {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsEmpty α] [IsEmpty β] : r ≃r s

Two relations on empty types are isomorphic.

Equations

• RelIso.relIsoOfIsEmpty r s = { toEquiv := Equiv.equivOfIsEmpty α β, map_rel_iff' := (_ : ∀ (a : α) {b : α}, s ((Equiv.equivOfIsEmpty α β) a) ((Equiv.equivOfIsEmpty α β) b) ↔ r a b) }

def RelIso.relIsoOfUniqueOfIrrefl {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsIrrefl α r] [IsIrrefl β s] [Unique α] [Unique β] : r ≃r s

Two irreflexive relations on a unique type are isomorphic.

Equations

• RelIso.relIsoOfUniqueOfIrrefl r s = { toEquiv := Equiv.equivOfUnique α β, map_rel_iff' := (_ : ∀ {a b : α}, s ((Equiv.equivOfUnique α β) a) ((Equiv.equivOfUnique α β) b) ↔ r a b) }

def RelIso.relIsoOfUniqueOfRefl {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsRefl α r] [IsRefl β s] [Unique α] [Unique β] : r ≃r s

Two reflexive relations on a unique type are isomorphic.

Equations

• RelIso.relIsoOfUniqueOfRefl r s = { toEquiv := Equiv.equivOfUnique α β, map_rel_iff' := (_ : ∀ {a b : α}, s ((Equiv.equivOfUnique α β) a) ((Equiv.equivOfUnique α β) b) ↔ r a b) }