order.rel_iso - mathlib docs

@[nolint]

structure rel_hom {α : Type u_4} {β : Type u_5} (r : α → α → Prop) (s : β → β → Prop) :

Type (max u_4 u_5)

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

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).

source

@[class]

structure rel_hom_class (F : Type u_4) {α : out_param (Type u_5)} {β : out_param (Type u_6)} (r : out_param (α → α → Prop)) (s : out_param (β → β → Prop)) :

Type (max u_4 u_5 u_6)

to_fun_like : fun_like F α (λ (_x : α), β)
map_rel : ∀ (f : F) {a b : α}, r a b → s (⇑f a) (⇑f b)

rel_hom_class 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 out_params since figuring them out from a goal is a higher-order matching problem that Lean usually can't do unaided.

Instances

source

theorem rel_hom_class.map_inf {α : Type u_1} {β : Type u_2} {F : Type u_4} [semilattice_inf α] [linear_order β] [rel_hom_class F has_lt.lt has_lt.lt] (a : F) (m n : β) :

⇑a (m ⊓ n) = ⇑a m ⊓ ⇑a n

source

theorem rel_hom_class.map_sup {α : Type u_1} {β : Type u_2} {F : Type u_4} [semilattice_sup α] [linear_order β] [rel_hom_class F gt gt] (a : F) (m n : β) :

⇑a (m ⊔ n) = ⇑a m ⊔ ⇑a n

source

@[protected]

theorem rel_hom_class.is_irrefl {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {F : Type u_4} [rel_hom_class F r s] (f : F) [is_irrefl β s] :

is_irrefl α r

source

@[protected]

theorem rel_hom_class.is_asymm {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {F : Type u_4} [rel_hom_class F r s] (f : F) [is_asymm β s] :

is_asymm α r

source

@[protected]

theorem rel_hom_class.acc {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {F : Type u_4} [rel_hom_class F r s] (f : F) (a : α) :

acc s (⇑f a) → acc r a

source

@[protected]

theorem rel_hom_class.well_founded {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {F : Type u_4} [rel_hom_class F r s] (f : F) (h : well_founded s) :

well_founded r

source

@[protected, instance]

def rel_hom.rel_hom_class {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} :

rel_hom_class (r →r s) r s

Equations

rel_hom.rel_hom_class = {to_fun_like := {coe := λ (o : r →r s), o.to_fun, coe_injective' := _}, map_rel := _}

source

@[protected, instance]

def rel_hom.has_coe_to_fun {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} :

has_coe_to_fun (r →r s) (λ (_x : r →r s), α → β)

Auxiliary instance if rel_hom_class.to_fun_like.to_has_coe_to_fun isn't found

Equations

rel_hom.has_coe_to_fun = {coe := λ (o : r →r s), o.to_fun}

source

@[protected]

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

r a b → s (⇑f a) (⇑f b)

source

@[simp]

theorem rel_hom.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}, r a b → s (f a) (f b)) :

⇑{to_fun := f, map_rel' := o} = f

source

@[simp]

theorem rel_hom.coe_fn_to_fun {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r →r s) :

f.to_fun = ⇑f

source

theorem rel_hom.coe_fn_injective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} :

function.injective coe_fn

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

source

@[ext]

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

f = g

source

theorem rel_hom.ext_iff {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {f g : r →r s} :

f = g ↔ ∀ (x : α), ⇑f x = ⇑g x

source

@[simp]

theorem rel_hom.id_apply {α : Type u_1} (r : α → α → Prop) (x : α) :

⇑(rel_hom.id r) x = x

source

@[protected]

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

r →r r

Identity map is a relation homomorphism.

Equations

rel_hom.id r = {to_fun := λ (x : α), x, map_rel' := _}

source

@[simp]

theorem rel_hom.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) :

⇑(g.comp f) x = ⇑g (⇑f x)

source

@[protected]

def rel_hom.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

g.comp f = {to_fun := λ (x : α), ⇑g (⇑f x), map_rel' := _}

source

@[protected]

def rel_hom.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

f.swap = {to_fun := ⇑f, map_rel' := _}

source

def rel_hom.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

rel_hom.preimage f s = {to_fun := f, map_rel' := _}

source

theorem injective_of_increasing {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [is_trichotomous α r] [is_irrefl 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

source

theorem rel_hom.injective_of_increasing {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [is_trichotomous α r] [is_irrefl β s] (f : r →r s) :

function.injective ⇑f

An increasing function is injective

source

theorem surjective.well_founded_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)) :

well_founded r ↔ well_founded s

source

structure rel_embedding {α : Type u_4} {β : Type u_5} (r : α → α → Prop) (s : β → β → Prop) :

Type (max u_4 u_5)

to_embedding : α ↪ β
map_rel_iff' : ∀ {a b : α}, s (⇑(self.to_embedding) a) (⇑(self.to_embedding) b) ↔ r a b

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).

source

def subtype.rel_embedding {X : Type u_1} (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

subtype.rel_embedding r p = {to_embedding := function.embedding.subtype p, map_rel_iff' := _}

source

theorem preimage_equivalence {α : Sort u_1} {β : Sort u_2} (f : α → β) {s : β → β → Prop} (hs : equivalence s) :

equivalence (f ⁻¹'o s)

source

def rel_embedding.to_rel_hom {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

f.to_rel_hom = {to_fun := f.to_embedding.to_fun, map_rel' := _}

source

@[protected, instance]

def rel_embedding.rel_hom.has_coe {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} :

has_coe (r ↪r s) (r →r s)

Equations

rel_embedding.rel_hom.has_coe = {coe := rel_embedding.to_rel_hom s}

source

@[protected, instance]

def rel_embedding.has_coe_to_fun {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} :

has_coe_to_fun (r ↪r s) (λ (_x : r ↪r s), α → β)

Equations

rel_embedding.has_coe_to_fun = {coe := λ (o : r ↪r s), ⇑(o.to_embedding)}

source

@[protected, instance]

def rel_embedding.rel_hom_class {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} :

rel_hom_class (r ↪r s) r s

Equations

rel_embedding.rel_hom_class = {to_fun_like := {coe := coe_fn rel_embedding.has_coe_to_fun, coe_injective' := _}, map_rel := _}

source

def rel_embedding.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 it is a composition of multiple projections.

Equations

rel_embedding.simps.apply h = ⇑h

source

@[simp]

theorem rel_embedding.to_rel_hom_eq_coe {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) :

f.to_rel_hom = ↑f

source

@[simp]

theorem rel_embedding.coe_coe_fn {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) :

⇑↑f = ⇑f

source

theorem rel_embedding.injective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) :

function.injective ⇑f

source

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

s (⇑f a) (⇑f b) ↔ r a b

source

@[simp]

theorem rel_embedding.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) :

⇑{to_embedding := f, map_rel_iff' := o} = ⇑f

source

@[simp]

theorem rel_embedding.coe_fn_to_embedding {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) :

⇑(f.to_embedding) = ⇑f

source

theorem rel_embedding.coe_fn_injective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} :

function.injective coe_fn

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

source

@[ext]

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

f = g

source

theorem rel_embedding.ext_iff {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {f g : r ↪r s} :

f = g ↔ ∀ (x : α), ⇑f x = ⇑g x

source

@[simp]

theorem rel_embedding.refl_apply {α : Type u_1} (r : α → α → Prop) (a : α) :

⇑(rel_embedding.refl r) a = a

source

@[protected]

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

r ↪r r

Identity map is a relation embedding.

Equations

rel_embedding.refl r = {to_embedding := function.embedding.refl α, map_rel_iff' := _}

source

@[protected]

def rel_embedding.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

f.trans g = {to_embedding := f.to_embedding.trans g.to_embedding, map_rel_iff' := _}

source

@[protected, instance]

def rel_embedding.inhabited {α : Type u_1} (r : α → α → Prop) :

inhabited (r ↪r r)

Equations

rel_embedding.inhabited r = {default := rel_embedding.refl r}

source

theorem rel_embedding.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) :

⇑(f.trans g) a = ⇑g (⇑f a)

source

@[simp]

theorem rel_embedding.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) :

⇑(f.trans g) = ⇑g ∘ ⇑f

source

@[protected]

def rel_embedding.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

f.swap = {to_embedding := f.to_embedding, map_rel_iff' := _}

source

def rel_embedding.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

rel_embedding.preimage f s = {to_embedding := f, map_rel_iff' := _}

source

theorem rel_embedding.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

source

@[protected]

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

is_irrefl α r

source

@[protected]

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

is_refl α r

source

@[protected]

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

is_symm α r

source

@[protected]

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

is_asymm α r

source

@[protected]

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

is_antisymm α r

source

@[protected]

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

is_trans α r

source

@[protected]

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

is_total α r

source

@[protected]

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

is_preorder α r

source

@[protected]

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

is_partial_order α r

source

@[protected]

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

is_linear_order α r

source

@[protected]

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

is_strict_order α r

source

@[protected]

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

is_trichotomous α r

source

@[protected]

theorem rel_embedding.is_strict_total_order' {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) [is_strict_total_order' β s] :

is_strict_total_order' α r

source

@[protected]

theorem rel_embedding.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

source

@[protected]

theorem rel_embedding.well_founded {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) (h : well_founded s) :

well_founded r

source

@[protected]

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

is_well_order α r

source

def rel_embedding.of_map_rel_iff {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : α → β) [is_antisymm α r] [is_refl β s] (hf : ∀ (a b : α), s (f a) (f b) ↔ r a b) :

r ↪r s

To define an 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

rel_embedding.of_map_rel_iff f hf = {to_embedding := {to_fun := f, inj' := _}, map_rel_iff' := hf}

source

@[simp]

theorem rel_embedding.of_map_rel_iff_coe {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : α → β) [is_antisymm α r] [is_refl β s] (hf : ∀ (a b : α), s (f a) (f b) ↔ r a b) :

⇑(rel_embedding.of_map_rel_iff f hf) = f

source

def rel_embedding.of_monotone {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [is_trichotomous α r] [is_asymm 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

rel_embedding.of_monotone f H = {to_embedding := {to_fun := f, inj' := _}, map_rel_iff' := _}

source

@[simp]

theorem rel_embedding.of_monotone_coe {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [is_trichotomous α r] [is_asymm β s] (f : α → β) (H : ∀ (a b : α), r a b → s (f a) (f b)) :

⇑(rel_embedding.of_monotone f H) = f

source

structure rel_iso {α : Type u_4} {β : Type u_5} (r : α → α → Prop) (s : β → β → Prop) :

Type (max u_4 u_5)

to_equiv : α ≃ β
map_rel_iff' : ∀ {a b : α}, s (⇑(self.to_equiv) a) (⇑(self.to_equiv) b) ↔ r a b

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

source

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

r ↪r s

Convert an rel_iso to an rel_embedding. This function is also available as a coercion but often it is easier to write f.to_rel_embedding than to write explicitly r and s in the target type.

Equations

f.to_rel_embedding = {to_embedding := f.to_equiv.to_embedding, map_rel_iff' := _}

source

theorem rel_iso.to_equiv_injective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} :

function.injective rel_iso.to_equiv

source

@[protected, instance]

def rel_iso.rel_embedding.has_coe {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} :

has_coe (r ≃r s) (r ↪r s)

Equations

rel_iso.rel_embedding.has_coe = {coe := rel_iso.to_rel_embedding s}

source

@[protected, instance]

def rel_iso.has_coe_to_fun {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} :

has_coe_to_fun (r ≃r s) (λ (_x : r ≃r s), α → β)

Equations

rel_iso.has_coe_to_fun = {coe := λ (f : r ≃r s), ⇑f}

source

@[protected, instance]

def rel_iso.rel_hom_class {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} :

rel_hom_class (r ≃r s) r s

Equations

rel_iso.rel_hom_class = {to_fun_like := {coe := coe_fn rel_iso.has_coe_to_fun, coe_injective' := _}, map_rel := _}

source

@[simp]

theorem rel_iso.to_rel_embedding_eq_coe {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) :

f.to_rel_embedding = ↑f

source

@[simp]

theorem rel_iso.coe_coe_fn {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) :

⇑↑f = ⇑f

source

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

s (⇑f a) (⇑f b) ↔ r a b

source

@[simp]

theorem rel_iso.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) :

⇑{to_equiv := f, map_rel_iff' := o} = ⇑f

source

@[simp]

theorem rel_iso.coe_fn_to_equiv {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) :

⇑(f.to_equiv) = ⇑f

source

theorem rel_iso.coe_fn_injective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} :

function.injective coe_fn

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.

source

@[ext]

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

f = g

source

theorem rel_iso.ext_iff {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {f g : r ≃r s} :

f = g ↔ ∀ (x : α), ⇑f x = ⇑g x

source

@[protected]

def rel_iso.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

f.symm = {to_equiv := f.to_equiv.symm, map_rel_iff' := _}

source

def rel_iso.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 it is a composition of multiple projections.

Equations

rel_iso.simps.apply h = ⇑h

source

def rel_iso.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

rel_iso.simps.symm_apply h = ⇑(h.symm)

source

@[protected]

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

r ≃r r

Identity map is a relation isomorphism.

Equations

rel_iso.refl r = {to_equiv := equiv.refl α, map_rel_iff' := _}

source

@[simp]

theorem rel_iso.refl_apply {α : Type u_1} (r : α → α → Prop) (a : α) :

⇑(rel_iso.refl r) a = a

source

@[protected]

def rel_iso.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

f₁.trans f₂ = {to_equiv := f₁.to_equiv.trans f₂.to_equiv, map_rel_iff' := _}

source

@[simp]

theorem rel_iso.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) (ᾰ : α) :

⇑(f₁.trans f₂) ᾰ = ⇑f₂ (⇑f₁ ᾰ)

source

@[protected, instance]

def rel_iso.inhabited {α : Type u_1} (r : α → α → Prop) :

inhabited (r ≃r r)

Equations

rel_iso.inhabited r = {default := rel_iso.refl r}

source

@[simp]

theorem rel_iso.default_def {α : Type u_1} (r : α → α → Prop) :

default = rel_iso.refl r

source

@[protected]

def rel_iso.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

f.swap = {to_equiv := f.to_equiv, map_rel_iff' := _}

source

@[simp]

theorem rel_iso.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) :

⇑({to_equiv := f, map_rel_iff' := o}.symm) = ⇑(f.symm)

source

@[simp]

theorem rel_iso.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 (⇑(e.symm) x) = x

source

@[simp]

theorem rel_iso.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) :

⇑(e.symm) (⇑e x) = x

source

theorem rel_iso.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 (⇑(e.symm) y) ↔ s (⇑e x) y

source

theorem rel_iso.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 (⇑(e.symm) x) y ↔ s x (⇑e y)

source

@[protected]

theorem rel_iso.bijective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) :

function.bijective ⇑e

source

@[protected]

theorem rel_iso.injective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) :

function.injective ⇑e

source

@[protected]

theorem rel_iso.surjective {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) :

function.surjective ⇑e

source

@[simp]

theorem rel_iso.range_eq {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) :

set.range ⇑e = set.univ

source

@[simp]

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

⇑f a = ⇑f b ↔ a = b

source

@[protected]

def rel_iso.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

rel_iso.preimage f s = {to_equiv := f, map_rel_iff' := _}

source

@[simp]

theorem rel_iso.of_surjective_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) (ᾰ : α) :

⇑(rel_iso.of_surjective f H) ᾰ = ⇑f ᾰ

source

noncomputable def rel_iso.of_surjective {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

rel_iso.of_surjective f H = {to_equiv := equiv.of_bijective ⇑f _, map_rel_iff' := _}

source

def rel_iso.sum_lex_congr {α₁ : Type u_1} {α₂ : Type u_2} {β₁ : Type u_3} {β₂ : Type u_4} {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

e₁.sum_lex_congr e₂ = {to_equiv := e₁.to_equiv.sum_congr e₂.to_equiv, map_rel_iff' := _}

source

def rel_iso.prod_lex_congr {α₁ : Type u_1} {α₂ : Type u_2} {β₁ : Type u_3} {β₂ : Type u_4} {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

e₁.prod_lex_congr e₂ = {to_equiv := e₁.to_equiv.prod_congr e₂.to_equiv, map_rel_iff' := _}

source

@[protected, instance]

def rel_iso.group {α : Type u_1} {r : α → α → Prop} :

group (r ≃r r)

Equations

rel_iso.group = {mul := λ (f₁ f₂ : r ≃r r), f₂.trans f₁, mul_assoc := _, one := rel_iso.refl r, one_mul := _, mul_one := _, npow := div_inv_monoid.npow._default (rel_iso.refl r) (λ (f₁ f₂ : r ≃r r), f₂.trans f₁) rel_iso.group._proof_4 rel_iso.group._proof_5, npow_zero' := _, npow_succ' := _, inv := rel_iso.symm r, div := div_inv_monoid.div._default (λ (f₁ f₂ : r ≃r r), f₂.trans f₁) rel_iso.group._proof_8 (rel_iso.refl r) rel_iso.group._proof_9 rel_iso.group._proof_10 (div_inv_monoid.npow._default (rel_iso.refl r) (λ (f₁ f₂ : r ≃r r), f₂.trans f₁) rel_iso.group._proof_4 rel_iso.group._proof_5) rel_iso.group._proof_11 rel_iso.group._proof_12 rel_iso.symm, div_eq_mul_inv := _, zpow := div_inv_monoid.zpow._default (λ (f₁ f₂ : r ≃r r), f₂.trans f₁) rel_iso.group._proof_14 (rel_iso.refl r) rel_iso.group._proof_15 rel_iso.group._proof_16 (div_inv_monoid.npow._default (rel_iso.refl r) (λ (f₁ f₂ : r ≃r r), f₂.trans f₁) rel_iso.group._proof_4 rel_iso.group._proof_5) rel_iso.group._proof_17 rel_iso.group._proof_18 rel_iso.symm, zpow_zero' := _, zpow_succ' := _, zpow_neg' := _, mul_left_inv := _}

source

@[simp]

theorem rel_iso.coe_one {α : Type u_1} {r : α → α → Prop} :

⇑1 = id

source

@[simp]

theorem rel_iso.coe_mul {α : Type u_1} {r : α → α → Prop} (e₁ e₂ : r ≃r r) :

⇑e₁ * e₂ = ⇑e₁ ∘ ⇑e₂

source

theorem rel_iso.mul_apply {α : Type u_1} {r : α → α → Prop} (e₁ e₂ : r ≃r r) (x : α) :

(⇑e₁ * e₂) x = ⇑e₁ (⇑e₂ x)

source

@[simp]

theorem rel_iso.inv_apply_self {α : Type u_1} {r : α → α → Prop} (e : r ≃r r) (x : α) :

⇑e⁻¹ (⇑e x) = x

source

@[simp]

theorem rel_iso.apply_inv_self {α : Type u_1} {r : α → α → Prop} (e : r ≃r r) (x : α) :

⇑e (⇑e⁻¹ x) = x

source

def subrel {α : Type u_1} (r : α → α → Prop) (p : set α) :

↥p → ↥p → Prop

subrel r p is the inherited relation on a subset.

Equations

subrel r p = coe ⁻¹'o r

source

@[simp]

theorem subrel_val {α : Type u_1} (r : α → α → Prop) (p : set α) {a b : ↥p} :

subrel r p a b ↔ r a.val b.val

source

@[protected]

def subrel.rel_embedding {α : Type u_1} (r : α → α → Prop) (p : set α) :

subrel r p ↪r r

The relation embedding from the inherited relation on a subset.

Equations

subrel.rel_embedding r p = {to_embedding := function.embedding.subtype (λ (x : α), x ∈ p), map_rel_iff' := _}

source

@[simp]

theorem subrel.rel_embedding_apply {α : Type u_1} (r : α → α → Prop) (p : set α) (a : ↥p) :

⇑(subrel.rel_embedding r p) a = a.val

source

@[protected, instance]

def subrel.is_well_order {α : Type u_1} (r : α → α → Prop) [is_well_order α r] (p : set α) :

is_well_order ↥p (subrel r p)

source

@[protected, instance]

def subrel.is_refl {α : Type u_1} (r : α → α → Prop) [is_refl α r] (p : set α) :

is_refl ↥p (subrel r p)

source

@[protected, instance]

def subrel.is_symm {α : Type u_1} (r : α → α → Prop) [is_symm α r] (p : set α) :

is_symm ↥p (subrel r p)

source

@[protected, instance]

def subrel.is_trans {α : Type u_1} (r : α → α → Prop) [is_trans α r] (p : set α) :

is_trans ↥p (subrel r p)

source

@[protected, instance]

def subrel.is_irrefl {α : Type u_1} (r : α → α → Prop) [is_irrefl α r] (p : set α) :

is_irrefl ↥p (subrel r p)

source

def rel_embedding.cod_restrict {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (p : set β) (f : r ↪r s) (H : ∀ (a : α), ⇑f a ∈ p) :

r ↪r subrel s p

Restrict the codomain of a relation embedding.

Equations

rel_embedding.cod_restrict p f H = {to_embedding := function.embedding.cod_restrict p f.to_embedding H, map_rel_iff' := _}

source

@[simp]

theorem rel_embedding.cod_restrict_apply {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (p : set β) (f : r ↪r s) (H : ∀ (a : α), ⇑f a ∈ p) (a : α) :

⇑(rel_embedding.cod_restrict p f H) a = ⟨⇑f a, _⟩

mathlib documentation

Relation homomorphisms, embeddings, isomorphisms #

Main declarations #

Notation #