Documentation

Mathlib.Order.Filter.Germ.Basic

Germ of a function at a filter #

The germ of a function f : α → β at a filter l : Filter α is the equivalence class of f with respect to the equivalence relation EventuallyEq l: f ≈ g means ∀ᶠ x in l, f x = g x.

Main definitions #

We define

We also define map (F : β → γ) : Germ l β → Germ l γ sending each germ f to F ∘ f.

For each of the following structures we prove that if β has this structure, then so does Germ l β:

Tags #

filter, germ

source
theorem Filter.const_eventuallyEq' {α : Type u_1} {β : Type u_2} {l : Filter α} [l.NeBot] {a : β} {b : β} :
(∀ᶠ (x : α) in l, a = b) ↔ a = b
source
theorem Filter.const_eventuallyEq {α : Type u_1} {β : Type u_2} {l : Filter α} [l.NeBot] {a : β} {b : β} :
((fun (x : α) => a) =ᶠ[l] fun (x : α) => b) ↔ a = b
source
def Filter.germSetoid {α : Type u_1} (l : Filter α) (β : Type u_5) :
Setoid (α → β)

Setoid used to define the space of germs.

Equations
  • l.germSetoid β = { r := l.EventuallyEq, iseqv := ⋯ }
source
def Filter.Germ {α : Type u_1} (l : Filter α) (β : Type u_5) :
Type (max u_1 u_5)

The space of germs of functions α → β at a filter l.

Equations
Instances For
source
def Filter.productSetoid {α : Type u_1} (l : Filter α) (ε : α → Type u_5) :
Setoid ((a : α) → ε a)

Setoid used to define the filter product. This is a dependent version of Filter.germSetoid.

Equations
  • l.productSetoid ε = { r := fun (f g : (a : α) → ε a) => ∀ᶠ (a : α) in l, f a = g a, iseqv := ⋯ }
source
def Filter.Product {α : Type u_1} (l : Filter α) (ε : α → Type u_5) :
Type (max u_1 u_5)

The filter product (a : α) → ε a at a filter l. This is a dependent version of Filter.Germ.

Equations
Instances For
source
instance Filter.Product.coeTC {α : Type u_1} {l : Filter α} {ε : α → Type u_5} :
CoeTC ((a : α) → ε a) (l.Product ε)
Equations
  • Filter.Product.coeTC = { coe := Quotient.mk' }
source
instance Filter.Product.instInhabited {α : Type u_1} {l : Filter α} {ε : α → Type u_5} [(a : α) → Inhabited (ε a)] :
Inhabited (l.Product ε)
Equations
  • Filter.Product.instInhabited = { default := Quotient.mk' fun (a : α) => default }
source
def Filter.Germ.ofFun {α : Type u_1} {β : Type u_2} {l : Filter α} :
(α → β) → l.Germ β
Equations
  • Filter.Germ.ofFun = Quotient.mk'
source
instance Filter.Germ.instCoeTCForall {α : Type u_1} {β : Type u_2} {l : Filter α} :
CoeTC (α → β) (l.Germ β)
Equations
  • Filter.Germ.instCoeTCForall = { coe := Filter.Germ.ofFun }
source
def Filter.Germ.const {α : Type u_1} {β : Type u_2} {l : Filter α} (b : β) :
l.Germ β
Equations
  • ↑b = ↑fun (x : α) => b
source
instance Filter.Germ.coeTC {α : Type u_1} {β : Type u_2} {l : Filter α} :
CoeTC β (l.Germ β)
Equations
  • Filter.Germ.coeTC = { coe := Filter.Germ.const }
source
def Filter.Germ.IsConstant {α : Type u_1} {β : Type u_2} {l : Filter α} (P : l.Germ β) :
Prop

A germ P of functions α → β is constant w.r.t. l.

Equations
source
theorem Filter.Germ.isConstant_coe {α : Type u_1} {β : Type u_2} {f : α → β} {l : Filter α} {b : β} (h : ∀ (x' : α), f x' = b) :
(↑f).IsConstant
source
@[simp]
theorem Filter.Germ.isConstant_coe_const {α : Type u_1} {β : Type u_2} {l : Filter α} {b : β} :
(↑fun (x : α) => b).IsConstant
source
theorem Filter.Germ.isConstant_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} {f : α → β} {g : β → γ} (h : (↑f).IsConstant) :
(↑(g ∘ f)).IsConstant

If f : α → β is constant w.r.t. l and g : β → γ, then g ∘ f : α → γ also is.

source
@[simp]
theorem Filter.Germ.quot_mk_eq_coe {α : Type u_1} {β : Type u_2} (l : Filter α) (f : α → β) :
Quot.mk Setoid.r f = ↑f
source
@[simp]
theorem Filter.Germ.mk'_eq_coe {α : Type u_1} {β : Type u_2} (l : Filter α) (f : α → β) :
Quotient.mk' f = ↑f
source
theorem Filter.Germ.inductionOn {α : Type u_1} {β : Type u_2} {l : Filter α} (f : l.Germ β) {p : l.Germ β → Prop} (h : ∀ (f : α → β), p ↑f) :
p f
source
theorem Filter.Germ.inductionOn₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (f : l.Germ β) (g : l.Germ γ) {p : l.Germ β → l.Germ γ → Prop} (h : ∀ (f : α → β) (g : α → γ), p ↑f ↑g) :
p f g
source
theorem Filter.Germ.inductionOn₃ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {l : Filter α} (f : l.Germ β) (g : l.Germ γ) (h : l.Germ δ) {p : l.Germ β → l.Germ γ → l.Germ δ → Prop} (H : ∀ (f : α → β) (g : α → γ) (h : α → δ), p ↑f ↑g ↑h) :
p f g h
source
def Filter.Germ.map' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {l : Filter α} {lc : Filter γ} (F : (α → β) → γ → δ) (hF : (l.EventuallyEq ⇒ lc.EventuallyEq) F F) :
l.Germ β → lc.Germ δ

Given a map F : (α → β) → (γ → δ) that sends functions eventually equal at l to functions eventually equal at lc, returns a map from Germ l β to Germ lc δ.

Equations
source
def Filter.Germ.liftOn {α : Type u_1} {β : Type u_2} {l : Filter α} {γ : Sort u_5} (f : l.Germ β) (F : (α → β) → γ) (hF : (l.EventuallyEq ⇒ fun (x1 x2 : γ) => x1 = x2) F F) :
γ

Given a germ f : Germ l β and a function F : (α → β) → γ sending eventually equal functions to the same value, returns the value F takes on functions having germ f at l.

Equations
source
@[simp]
theorem Filter.Germ.map'_coe {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {l : Filter α} {lc : Filter γ} (F : (α → β) → γ → δ) (hF : (l.EventuallyEq ⇒ lc.EventuallyEq) F F) (f : α → β) :
Filter.Germ.map' F hF ↑f = ↑(F f)
source
@[simp]
theorem Filter.Germ.coe_eq {α : Type u_1} {β : Type u_2} {l : Filter α} {f : α → β} {g : α → β} :
↑f = ↑g ↔ f =ᶠ[l] g
source
theorem Filter.EventuallyEq.germ_eq {α : Type u_1} {β : Type u_2} {l : Filter α} {f : α → β} {g : α → β} :
f =ᶠ[l] g → ↑f = ↑g

Alias of the reverse direction of Filter.Germ.coe_eq.

source
def Filter.Germ.map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (op : β → γ) :
l.Germ β → l.Germ γ

Lift a function β → γ to a function Germ l β → Germ l γ.

Equations
source
@[simp]
theorem Filter.Germ.map_coe {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (op : β → γ) (f : α → β) :
Filter.Germ.map op ↑f = ↑(op ∘ f)
source
@[simp]
theorem Filter.Germ.map_id {α : Type u_1} {β : Type u_2} {l : Filter α} :
Filter.Germ.map id = id
source
theorem Filter.Germ.map_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {l : Filter α} (op₁ : γ → δ) (op₂ : β → γ) (f : l.Germ β) :
Filter.Germ.map op₁ (Filter.Germ.map op₂ f) = Filter.Germ.map (op₁ ∘ op₂) f
source
def Filter.Germ.map₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {l : Filter α} (op : β → γ → δ) :
l.Germ β → l.Germ γ → l.Germ δ

Lift a binary function β → γ → δ to a function Germ l β → Germ l γ → Germ l δ.

Equations
source
@[simp]
theorem Filter.Germ.map₂_coe {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {l : Filter α} (op : β → γ → δ) (f : α → β) (g : α → γ) :
Filter.Germ.map₂ op ↑f ↑g = ↑fun (x : α) => op (f x) (g x)
source
def Filter.Germ.Tendsto {α : Type u_1} {β : Type u_2} {l : Filter α} (f : l.Germ β) (lb : Filter β) :
Prop

A germ at l of maps from α to β tends to lb : Filter β if it is represented by a map which tends to lb along l.

Equations
source
@[simp]
theorem Filter.Germ.coe_tendsto {α : Type u_1} {β : Type u_2} {l : Filter α} {f : α → β} {lb : Filter β} :
(↑f).Tendsto lb ↔ Filter.Tendsto f l lb
source
theorem Filter.Tendsto.germ_tendsto {α : Type u_1} {β : Type u_2} {l : Filter α} {f : α → β} {lb : Filter β} :
Filter.Tendsto f l lb → (↑f).Tendsto lb

Alias of the reverse direction of Filter.Germ.coe_tendsto.

source
def Filter.Germ.compTendsto' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (f : l.Germ β) {lc : Filter γ} (g : lc.Germ α) (hg : g.Tendsto l) :
lc.Germ β

Given two germs f : Germ l β, and g : Germ lc α, where l : Filter α, if g tends to l, then the composition f ∘ g is well-defined as a germ at lc.

Equations
source
@[simp]
theorem Filter.Germ.coe_compTendsto' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (f : α → β) {lc : Filter γ} {g : lc.Germ α} (hg : g.Tendsto l) :
(↑f).compTendsto' g hg = Filter.Germ.map f g
source
def Filter.Germ.compTendsto {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (f : l.Germ β) {lc : Filter γ} (g : γ → α) (hg : Filter.Tendsto g lc l) :
lc.Germ β

Given a germ f : Germ l β and a function g : γ → α, where l : Filter α, if g tends to l along lc : Filter γ, then the composition f ∘ g is well-defined as a germ at lc.

Equations
  • f.compTendsto g hg = f.compTendsto' ↑g ⋯
source
@[simp]
theorem Filter.Germ.coe_compTendsto {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (f : α → β) {lc : Filter γ} {g : γ → α} (hg : Filter.Tendsto g lc l) :
(↑f).compTendsto g hg = ↑(f ∘ g)
source
@[simp]
theorem Filter.Germ.compTendsto'_coe {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (f : l.Germ β) {lc : Filter γ} {g : γ → α} (hg : Filter.Tendsto g lc l) :
f.compTendsto' ↑g ⋯ = f.compTendsto g hg
source
theorem Filter.Germ.Filter.Tendsto.congr_germ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : β → γ} {g : β → γ} {l : Filter α} {l' : Filter β} (h : f =ᶠ[l'] g) {φ : α → β} (hφ : Filter.Tendsto φ l l') :
↑(f ∘ φ) = ↑(g ∘ φ)
source
theorem Filter.Germ.isConstant_comp_tendsto {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} {f : α → β} {lc : Filter γ} {g : γ → α} (hf : (↑f).IsConstant) (hg : Filter.Tendsto g lc l) :
(↑(f ∘ g)).IsConstant
source
theorem Filter.Germ.isConstant_compTendsto {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} {f : l.Germ β} {lc : Filter γ} {g : γ → α} (hf : f.IsConstant) (hg : Filter.Tendsto g lc l) :
(f.compTendsto g hg).IsConstant

If a germ f : Germ l β is constant, where l : Filter α, and a function g : γ → α tends to l along lc : Filter γ, the germ of the composition f ∘ g is also constant.

source
@[simp]
theorem Filter.Germ.const_inj {α : Type u_1} {β : Type u_2} {l : Filter α} [l.NeBot] {a : β} {b : β} :
↑a = ↑b ↔ a = b
source
@[simp]
theorem Filter.Germ.map_const {α : Type u_1} {β : Type u_2} {γ : Type u_3} (l : Filter α) (a : β) (f : β → γ) :
Filter.Germ.map f ↑a = ↑(f a)
source
@[simp]
theorem Filter.Germ.map₂_const {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (l : Filter α) (b : β) (c : γ) (f : β → γ → δ) :
Filter.Germ.map₂ f ↑b ↑c = ↑(f b c)
source
@[simp]
theorem Filter.Germ.const_compTendsto {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (b : β) {lc : Filter γ} {g : γ → α} (hg : Filter.Tendsto g lc l) :
(↑b).compTendsto g hg = ↑b
source
@[simp]
theorem Filter.Germ.const_compTendsto' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (b : β) {lc : Filter γ} {g : lc.Germ α} (hg : g.Tendsto l) :
(↑b).compTendsto' g hg = ↑b
source
def Filter.Germ.LiftPred {α : Type u_1} {β : Type u_2} {l : Filter α} (p : β → Prop) (f : l.Germ β) :
Prop

Lift a predicate on β to Germ l β.

Equations
source
@[simp]
theorem Filter.Germ.liftPred_coe {α : Type u_1} {β : Type u_2} {l : Filter α} {p : β → Prop} {f : α → β} :
Filter.Germ.LiftPred p ↑f ↔ ∀ᶠ (x : α) in l, p (f x)
source
theorem Filter.Germ.liftPred_const {α : Type u_1} {β : Type u_2} {l : Filter α} {p : β → Prop} {x : β} (hx : p x) :
Filter.Germ.LiftPred p ↑x
source
@[simp]
theorem Filter.Germ.liftPred_const_iff {α : Type u_1} {β : Type u_2} {l : Filter α} [l.NeBot] {p : β → Prop} {x : β} :
Filter.Germ.LiftPred p ↑x ↔ p x
source
def Filter.Germ.LiftRel {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (r : β → γ → Prop) (f : l.Germ β) (g : l.Germ γ) :
Prop

Lift a relation r : β → γ → Prop to Germ l β → Germ l γ → Prop.

Equations
source
@[simp]
theorem Filter.Germ.liftRel_coe {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} {r : β → γ → Prop} {f : α → β} {g : α → γ} :
Filter.Germ.LiftRel r ↑f ↑g ↔ ∀ᶠ (x : α) in l, r (f x) (g x)
source
theorem Filter.Germ.liftRel_const {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} {r : β → γ → Prop} {x : β} {y : γ} (h : r x y) :
Filter.Germ.LiftRel r ↑x ↑y
source
@[simp]
theorem Filter.Germ.liftRel_const_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} [l.NeBot] {r : β → γ → Prop} {x : β} {y : γ} :
Filter.Germ.LiftRel r ↑x ↑y ↔ r x y
source
instance Filter.Germ.instInhabited {α : Type u_1} {β : Type u_2} {l : Filter α} [Inhabited β] :
Inhabited (l.Germ β)
Equations
  • Filter.Germ.instInhabited = { default := ↑default }
source
instance Filter.Germ.instAdd {α : Type u_1} {l : Filter α} {M : Type u_5} [Add M] :
Add (l.Germ M)
Equations
source
instance Filter.Germ.instMul {α : Type u_1} {l : Filter α} {M : Type u_5} [Mul M] :
Mul (l.Germ M)
Equations
source
@[simp]
theorem Filter.Germ.coe_add {α : Type u_1} {l : Filter α} {M : Type u_5} [Add M] (f : α → M) (g : α → M) :
↑(f + g) = ↑f + ↑g
source
@[simp]
theorem Filter.Germ.coe_mul {α : Type u_1} {l : Filter α} {M : Type u_5} [Mul M] (f : α → M) (g : α → M) :
↑(f * g) = ↑f * ↑g
source
instance Filter.Germ.instZero {α : Type u_1} {l : Filter α} {M : Type u_5} [Zero M] :
Zero (l.Germ M)
Equations
  • Filter.Germ.instZero = { zero := ↑0 }
source
instance Filter.Germ.instOne {α : Type u_1} {l : Filter α} {M : Type u_5} [One M] :
One (l.Germ M)
Equations
  • Filter.Germ.instOne = { one := ↑1 }
source
@[simp]
theorem Filter.Germ.coe_zero {α : Type u_1} {l : Filter α} {M : Type u_5} [Zero M] :
↑0 = 0
source
@[simp]
theorem Filter.Germ.coe_one {α : Type u_1} {l : Filter α} {M : Type u_5} [One M] :
↑1 = 1
source
instance Filter.Germ.instAddSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [AddSemigroup M] :
AddSemigroup (l.Germ M)
Equations
source
theorem Filter.Germ.instAddSemigroup.proof_1 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddSemigroup M] (a : l.Germ M) (b : l.Germ M) (c : l.Germ M) :
a + b + c = a + (b + c)
source
instance Filter.Germ.instSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [Semigroup M] :
Semigroup (l.Germ M)
Equations
source
theorem Filter.Germ.instAddCommSemigroup.proof_1 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddCommSemigroup M] (q₁ : Quotient (l.germSetoid M)) (q₂ : Quotient (l.germSetoid M)) :
q₁ + q₂ = q₂ + q₁
source
instance Filter.Germ.instAddCommSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [AddCommSemigroup M] :
AddCommSemigroup (l.Germ M)
Equations
source
instance Filter.Germ.instCommSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [CommSemigroup M] :
CommSemigroup (l.Germ M)
Equations
source
instance Filter.Germ.instIsAddLeftCancel {α : Type u_1} {l : Filter α} {M : Type u_5} [Add M] [IsLeftCancelAdd M] :
IsLeftCancelAdd (l.Germ M)
Equations
  • ⋯ = ⋯
source
instance Filter.Germ.instIsLeftCancelMul {α : Type u_1} {l : Filter α} {M : Type u_5} [Mul M] [IsLeftCancelMul M] :
IsLeftCancelMul (l.Germ M)
Equations
  • ⋯ = ⋯
source
instance Filter.Germ.instIsAddRightCancel {α : Type u_1} {l : Filter α} {M : Type u_5} [Add M] [IsRightCancelAdd M] :
IsRightCancelAdd (l.Germ M)
Equations
  • ⋯ = ⋯
source
instance Filter.Germ.instIsRightCancelMul {α : Type u_1} {l : Filter α} {M : Type u_5} [Mul M] [IsRightCancelMul M] :
IsRightCancelMul (l.Germ M)
Equations
  • ⋯ = ⋯
source
instance Filter.Germ.instIsAddCancel {α : Type u_1} {l : Filter α} {M : Type u_5} [Add M] [IsCancelAdd M] :
IsCancelAdd (l.Germ M)
Equations
  • ⋯ = ⋯
source
instance Filter.Germ.instIsCancelMul {α : Type u_1} {l : Filter α} {M : Type u_5} [Mul M] [IsCancelMul M] :
IsCancelMul (l.Germ M)
Equations
  • ⋯ = ⋯
source
instance Filter.Germ.instAddLeftCancelSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [AddLeftCancelSemigroup M] :
AddLeftCancelSemigroup (l.Germ M)
Equations
source
theorem Filter.Germ.instAddLeftCancelSemigroup.proof_1 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddLeftCancelSemigroup M] :
∀ (x x_1 x_2 : l.Germ M), x + x_1 = x + x_2 → x_1 = x_2
source
instance Filter.Germ.instLeftCancelSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [LeftCancelSemigroup M] :
LeftCancelSemigroup (l.Germ M)
Equations
source
instance Filter.Germ.instAddRightCancelSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [AddRightCancelSemigroup M] :
AddRightCancelSemigroup (l.Germ M)
Equations
source
theorem Filter.Germ.instAddRightCancelSemigroup.proof_1 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddRightCancelSemigroup M] :
∀ (x x_1 x_2 : l.Germ M), x + x_1 = x_2 + x_1 → x = x_2
source
instance Filter.Germ.instRightCancelSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [RightCancelSemigroup M] :
RightCancelSemigroup (l.Germ M)
Equations
source
theorem Filter.Germ.instAddZeroClass.proof_2 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddZeroClass M] (q : Quotient (l.germSetoid M)) :
q + 0 = q
source
instance Filter.Germ.instAddZeroClass {α : Type u_1} {l : Filter α} {M : Type u_5} [AddZeroClass M] :
AddZeroClass (l.Germ M)
Equations
source
theorem Filter.Germ.instAddZeroClass.proof_1 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddZeroClass M] (q : Quotient (l.germSetoid M)) :
0 + q = q
source
instance Filter.Germ.instMulOneClass {α : Type u_1} {l : Filter α} {M : Type u_5} [MulOneClass M] :
MulOneClass (l.Germ M)
Equations
source
instance Filter.Germ.instVAdd {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [VAdd M G] :
VAdd M (l.Germ G)
Equations
source
instance Filter.Germ.instSMul {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [SMul M G] :
SMul M (l.Germ G)
Equations
source
instance Filter.Germ.instPow {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [Pow G M] :
Pow (l.Germ G) M
Equations
  • Filter.Germ.instPow = { pow := fun (f : l.Germ G) (n : M) => Filter.Germ.map (fun (x : G) => x ^ n) f }
source
@[simp]
theorem Filter.Germ.coe_vadd {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [VAdd M G] (n : M) (f : α → G) :
↑(n +ᵥ f) = n +ᵥ ↑f
source
@[simp]
theorem Filter.Germ.coe_smul {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [SMul M G] (n : M) (f : α → G) :
↑(n • f) = n • ↑f
source
@[simp]
theorem Filter.Germ.const_vadd {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [VAdd M G] (n : M) (a : G) :
↑(n +ᵥ a) = n +ᵥ ↑a
source
@[simp]
theorem Filter.Germ.const_smul {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [SMul M G] (n : M) (a : G) :
↑(n • a) = n • ↑a
source
@[simp]
theorem Filter.Germ.coe_nsmul {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [SMul M G] (f : α → G) (n : M) :
↑(n • f) = n • ↑f
source
@[simp]
theorem Filter.Germ.coe_pow {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [Pow G M] (f : α → G) (n : M) :
↑(f ^ n) = ↑f ^ n
source
@[simp]
theorem Filter.Germ.const_nsmul {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [SMul M G] (a : G) (n : M) :
↑(n • a) = n • ↑a
source
@[simp]
theorem Filter.Germ.const_pow {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [Pow G M] (a : G) (n : M) :
↑(a ^ n) = ↑a ^ n
source
theorem Filter.Germ.instAddMonoid.proof_7 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddMonoid M] (x : l.Germ M) :
AddMonoid.nsmul 0 x = 0
source
theorem Filter.Germ.instAddMonoid.proof_6 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddMonoid M] (a : l.Germ M) :
a + 0 = a
source
instance Filter.Germ.instAddMonoid {α : Type u_1} {l : Filter α} {M : Type u_5} [AddMonoid M] :
AddMonoid (l.Germ M)
Equations
  • Filter.Germ.instAddMonoid = AddMonoid.mk ⋯ ⋯ (fun (n : ℕ) (a : l.Germ M) => n • a) ⋯ ⋯
source
theorem Filter.Germ.instAddMonoid.proof_2 {α : Type u_2} {l : Filter α} {M : Type u_1} [AddMonoid M] :
↑0 = ↑0
source
theorem Filter.Germ.instAddMonoid.proof_1 {α : Type u_2} {l : Filter α} {M : Type u_1} :
Function.Surjective (Quot.mk Setoid.r)
source
theorem Filter.Germ.instAddMonoid.proof_8 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddMonoid M] (n : ℕ) (x : l.Germ M) :
AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
source
theorem Filter.Germ.instAddMonoid.proof_4 {α : Type u_2} {l : Filter α} {M : Type u_1} [AddMonoid M] :
∀ (x : α → M) (x_1 : ℕ), ↑(x_1 • x) = ↑(x_1 • x)
source
theorem Filter.Germ.instAddMonoid.proof_3 {α : Type u_2} {l : Filter α} {M : Type u_1} [AddMonoid M] :
∀ (x x_1 : α → M), ↑(x + x_1) = ↑(x + x_1)
source
theorem Filter.Germ.instAddMonoid.proof_5 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddMonoid M] (a : l.Germ M) :
0 + a = a
source
instance Filter.Germ.instMonoid {α : Type u_1} {l : Filter α} {M : Type u_5} [Monoid M] :
Monoid (l.Germ M)
Equations
  • Filter.Germ.instMonoid = Monoid.mk ⋯ ⋯ (fun (n : ℕ) (a : l.Germ M) => a ^ n) ⋯ ⋯
source
theorem Filter.Germ.coeAddHom.proof_2 {α : Type u_1} {M : Type u_2} [AddMonoid M] (l : Filter α) :
∀ (x x_1 : α → M), { toFun := Filter.Germ.ofFun, map_zero' := ⋯ }.toFun (x + x_1) = { toFun := Filter.Germ.ofFun, map_zero' := ⋯ }.toFun (x + x_1)
source
theorem Filter.Germ.coeAddHom.proof_1 {α : Type u_1} {M : Type u_2} [AddMonoid M] (l : Filter α) :
↑0 = ↑0
source
def Filter.Germ.coeAddHom {α : Type u_1} {M : Type u_5} [AddMonoid M] (l : Filter α) :
(α → M) →+ l.Germ M

Coercion from functions to germs as an additive monoid homomorphism.

Equations
source
def Filter.Germ.coeMulHom {α : Type u_1} {M : Type u_5} [Monoid M] (l : Filter α) :
(α → M) →* l.Germ M

Coercion from functions to germs as a monoid homomorphism.

Equations
source
@[simp]
theorem Filter.Germ.coe_coeAddHom {α : Type u_1} {l : Filter α} {M : Type u_5} [AddMonoid M] :
⇑(Filter.Germ.coeAddHom l) = Filter.Germ.ofFun
source
@[simp]
theorem Filter.Germ.coe_coeMulHom {α : Type u_1} {l : Filter α} {M : Type u_5} [Monoid M] :
⇑(Filter.Germ.coeMulHom l) = Filter.Germ.ofFun
source
instance Filter.Germ.instAddCommMonoid {α : Type u_1} {l : Filter α} {M : Type u_5} [AddCommMonoid M] :
AddCommMonoid (l.Germ M)
Equations
source
theorem Filter.Germ.instAddCommMonoid.proof_1 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddCommMonoid M] (a : l.Germ M) (b : l.Germ M) :
a + b = b + a
source
instance Filter.Germ.instCommMonoid {α : Type u_1} {l : Filter α} {M : Type u_5} [CommMonoid M] :
CommMonoid (l.Germ M)
Equations
source
instance Filter.Germ.instNatCast {α : Type u_1} {l : Filter α} {M : Type u_5} [NatCast M] :
NatCast (l.Germ M)
Equations
  • Filter.Germ.instNatCast = { natCast := fun (n : ℕ) => ↑↑n }
source
@[simp]
theorem Filter.Germ.natCast_def {α : Type u_1} {l : Filter α} {M : Type u_5} [NatCast M] (n : ℕ) :
(↑fun (x : α) => ↑n) = ↑n
source
@[simp]
theorem Filter.Germ.const_nat {α : Type u_1} {l : Filter α} {M : Type u_5} [NatCast M] (n : ℕ) :
↑↑n = ↑n
source
@[simp]
theorem Filter.Germ.coe_ofNat {α : Type u_1} {l : Filter α} {M : Type u_5} [NatCast M] (n : ℕ) [n.AtLeastTwo] :
↑(OfNat.ofNat n) = OfNat.ofNat n
source
@[simp]
theorem Filter.Germ.const_ofNat {α : Type u_1} {l : Filter α} {M : Type u_5} [NatCast M] (n : ℕ) [n.AtLeastTwo] :
↑(OfNat.ofNat n) = OfNat.ofNat n
source
instance Filter.Germ.instIntCast {α : Type u_1} {l : Filter α} {M : Type u_5} [IntCast M] :
IntCast (l.Germ M)
Equations
  • Filter.Germ.instIntCast = { intCast := fun (n : ℤ) => ↑↑n }
source
@[simp]
theorem Filter.Germ.intCast_def {α : Type u_1} {l : Filter α} {M : Type u_5} [IntCast M] (n : ℤ) :
(↑fun (x : α) => ↑n) = ↑n
source
@[deprecated Filter.Germ.natCast_def]
theorem Filter.Germ.coe_nat {α : Type u_1} {l : Filter α} {M : Type u_5} [NatCast M] (n : ℕ) :
(↑fun (x : α) => ↑n) = ↑n

Alias of Filter.Germ.natCast_def.

source
@[deprecated Filter.Germ.intCast_def]
theorem Filter.Germ.coe_int {α : Type u_1} {l : Filter α} {M : Type u_5} [IntCast M] (n : ℤ) :
(↑fun (x : α) => ↑n) = ↑n

Alias of Filter.Germ.intCast_def.

source
instance Filter.Germ.instAddMonoidWithOne {α : Type u_1} {l : Filter α} {M : Type u_5} [AddMonoidWithOne M] :
AddMonoidWithOne (l.Germ M)
Equations
source
instance Filter.Germ.instAddCommMonoidWithOne {α : Type u_1} {l : Filter α} {M : Type u_5} [AddCommMonoidWithOne M] :
AddCommMonoidWithOne (l.Germ M)
Equations
source
instance Filter.Germ.instNeg {α : Type u_1} {l : Filter α} {G : Type u_6} [Neg G] :
Neg (l.Germ G)
Equations
source
instance Filter.Germ.instInv {α : Type u_1} {l : Filter α} {G : Type u_6} [Inv G] :
Inv (l.Germ G)
Equations
source
@[simp]
theorem Filter.Germ.coe_neg {α : Type u_1} {l : Filter α} {G : Type u_6} [Neg G] (f : α → G) :
↑(-f) = -↑f
source
@[simp]
theorem Filter.Germ.coe_inv {α : Type u_1} {l : Filter α} {G : Type u_6} [Inv G] (f : α → G) :
↑f⁻¹ = (↑f)⁻¹
source
@[simp]
theorem Filter.Germ.const_neg {α : Type u_1} {l : Filter α} {G : Type u_6} [Neg G] (a : G) :
↑(-a) = -↑a
source
@[simp]
theorem Filter.Germ.const_inv {α : Type u_1} {l : Filter α} {G : Type u_6} [Inv G] (a : G) :
↑a⁻¹ = (↑a)⁻¹
source
instance Filter.Germ.instSub {α : Type u_1} {l : Filter α} {M : Type u_5} [Sub M] :
Sub (l.Germ M)
Equations
source
instance Filter.Germ.instDiv {α : Type u_1} {l : Filter α} {M : Type u_5} [Div M] :
Div (l.Germ M)
Equations
source
@[simp]
theorem Filter.Germ.coe_sub {α : Type u_1} {l : Filter α} {M : Type u_5} [Sub M] (f : α → M) (g : α → M) :
↑(f - g) = ↑f - ↑g
source
@[simp]
theorem Filter.Germ.coe_div {α : Type u_1} {l : Filter α} {M : Type u_5} [Div M] (f : α → M) (g : α → M) :
↑(f / g) = ↑f / ↑g
source
@[simp]
theorem Filter.Germ.const_sub {α : Type u_1} {l : Filter α} {M : Type u_5} [Sub M] (a : M) (b : M) :
↑(a - b) = ↑a - ↑b
source
@[simp]
theorem Filter.Germ.const_div {α : Type u_1} {l : Filter α} {M : Type u_5} [Div M] (a : M) (b : M) :
↑(a / b) = ↑a / ↑b
source
theorem Filter.Germ.instInvolutiveNeg.proof_1 {α : Type u_1} {l : Filter α} {G : Type u_2} [InvolutiveNeg G] (q : Quotient (l.germSetoid G)) :
- -q = q
source
instance Filter.Germ.instInvolutiveNeg {α : Type u_1} {l : Filter α} {G : Type u_6} [InvolutiveNeg G] :
InvolutiveNeg (l.Germ G)
Equations
source
instance Filter.Germ.instInvolutiveInv {α : Type u_1} {l : Filter α} {G : Type u_6} [InvolutiveInv G] :
InvolutiveInv (l.Germ G)
Equations
source
instance Filter.Germ.instHasDistribNeg {α : Type u_1} {l : Filter α} {G : Type u_6} [Mul G] [HasDistribNeg G] :
HasDistribNeg (l.Germ G)
Equations
source
theorem Filter.Germ.instNegZeroClass.proof_1 {α : Type u_2} {l : Filter α} {G : Type u_1} [NegZeroClass G] :
↑((fun (x : α → G) => Neg.neg ∘ x) fun (x : α) => 0) = ↑fun (x : α) => 0
source
instance Filter.Germ.instNegZeroClass {α : Type u_1} {l : Filter α} {G : Type u_6} [NegZeroClass G] :
NegZeroClass (l.Germ G)
Equations
source
instance Filter.Germ.instInvOneClass {α : Type u_1} {l : Filter α} {G : Type u_6} [InvOneClass G] :
InvOneClass (l.Germ G)
Equations
source
instance Filter.Germ.subNegMonoid {α : Type u_1} {l : Filter α} {G : Type u_6} [SubNegMonoid G] :
SubNegMonoid (l.Germ G)
Equations
source
theorem Filter.Germ.subNegMonoid.proof_2 {α : Type u_1} {l : Filter α} {G : Type u_2} [SubNegMonoid G] (q : Quotient (l.germSetoid G)) :
(fun (z : ℤ) (f : l.Germ G) => z • f) 0 q = 0
source
theorem Filter.Germ.subNegMonoid.proof_1 {α : Type u_1} {l : Filter α} {G : Type u_2} [SubNegMonoid G] (q₁ : Quotient (l.germSetoid G)) (q₂ : Quotient (l.germSetoid G)) :
q₁ - q₂ = q₁ + -q₂
source
theorem Filter.Germ.subNegMonoid.proof_4 {α : Type u_1} {l : Filter α} {G : Type u_2} [SubNegMonoid G] :
∀ (x : ℕ) (q : Quotient (l.germSetoid G)), (fun (z : ℤ) (f : l.Germ G) => z • f) (Int.negSucc x) q = -(fun (z : ℤ) (f : l.Germ G) => z • f) (↑x.succ) q
source
theorem Filter.Germ.subNegMonoid.proof_3 {α : Type u_1} {l : Filter α} {G : Type u_2} [SubNegMonoid G] :
∀ (x : ℕ) (q : Quotient (l.germSetoid G)), (fun (z : ℤ) (f : l.Germ G) => z • f) (Int.ofNat x.succ) q = (fun (z : ℤ) (f : l.Germ G) => z • f) (Int.ofNat x) q + q
source
instance Filter.Germ.instDivInvMonoid {α : Type u_1} {l : Filter α} {G : Type u_6} [DivInvMonoid G] :
DivInvMonoid (l.Germ G)
Equations
  • Filter.Germ.instDivInvMonoid = DivInvMonoid.mk ⋯ (fun (z : ℤ) (f : l.Germ G) => f ^ z) ⋯ ⋯ ⋯
source
instance Filter.Germ.instSubtractionMonoid {α : Type u_1} {l : Filter α} {G : Type u_6} [SubtractionMonoid G] :
SubtractionMonoid (l.Germ G)
Equations
source
theorem Filter.Germ.instSubtractionMonoid.proof_2 {α : Type u_1} {l : Filter α} {G : Type u_2} [SubtractionMonoid G] (x : l.Germ G) (y : l.Germ G) :
-(x + y) = -y + -x
source
theorem Filter.Germ.instSubtractionMonoid.proof_1 {α : Type u_1} {l : Filter α} {G : Type u_2} [SubtractionMonoid G] (a : l.Germ G) :
- -a = a
source
theorem Filter.Germ.instSubtractionMonoid.proof_3 {α : Type u_1} {l : Filter α} {G : Type u_2} [SubtractionMonoid G] (x : l.Germ G) (y : l.Germ G) :
x + y = 0 → -x = y
source
instance Filter.Germ.instDivisionMonoid {α : Type u_1} {l : Filter α} {G : Type u_6} [DivisionMonoid G] :
DivisionMonoid (l.Germ G)
Equations
source
instance Filter.Germ.instAddGroup {α : Type u_1} {l : Filter α} {G : Type u_6} [AddGroup G] :
AddGroup (l.Germ G)
Equations
source
theorem Filter.Germ.instAddGroup.proof_1 {α : Type u_1} {l : Filter α} {G : Type u_2} [AddGroup G] (q : Quotient (l.germSetoid G)) :
-q + q = 0
source
instance Filter.Germ.instGroup {α : Type u_1} {l : Filter α} {G : Type u_6} [Group G] :
Group (l.Germ G)
Equations
source
theorem Filter.Germ.instAddCommGroup.proof_1 {α : Type u_1} {l : Filter α} {G : Type u_2} [AddCommGroup G] (a : l.Germ G) (b : l.Germ G) :
a + b = b + a
source
instance Filter.Germ.instAddCommGroup {α : Type u_1} {l : Filter α} {G : Type u_6} [AddCommGroup G] :
AddCommGroup (l.Germ G)
Equations
source
instance Filter.Germ.instCommGroup {α : Type u_1} {l : Filter α} {G : Type u_6} [CommGroup G] :
CommGroup (l.Germ G)
Equations
source
instance Filter.Germ.instAddGroupWithOne {α : Type u_1} {l : Filter α} {G : Type u_6} [AddGroupWithOne G] :
AddGroupWithOne (l.Germ G)
Equations
  • Filter.Germ.instAddGroupWithOne = AddGroupWithOne.mk ⋯ SubNegMonoid.zsmul ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
source
instance Filter.Germ.instNontrivial {α : Type u_1} {l : Filter α} {R : Type u_5} [Nontrivial R] [l.NeBot] :
Nontrivial (l.Germ R)
Equations
  • ⋯ = ⋯
source
instance Filter.Germ.instMulZeroClass {α : Type u_1} {l : Filter α} {R : Type u_5} [MulZeroClass R] :
MulZeroClass (l.Germ R)
Equations
source
instance Filter.Germ.instMulZeroOneClass {α : Type u_1} {l : Filter α} {R : Type u_5} [MulZeroOneClass R] :
MulZeroOneClass (l.Germ R)
Equations
source
instance Filter.Germ.instMonoidWithZero {α : Type u_1} {l : Filter α} {R : Type u_5} [MonoidWithZero R] :
MonoidWithZero (l.Germ R)
Equations
source
instance Filter.Germ.instDistrib {α : Type u_1} {l : Filter α} {R : Type u_5} [Distrib R] :
Distrib (l.Germ R)
Equations
source
instance Filter.Germ.instNonUnitalNonAssocSemiring {α : Type u_1} {l : Filter α} {R : Type u_5} [NonUnitalNonAssocSemiring R] :
NonUnitalNonAssocSemiring (l.Germ R)
Equations
source
instance Filter.Germ.instNonUnitalSemiring {α : Type u_1} {l : Filter α} {R : Type u_5} [NonUnitalSemiring R] :
NonUnitalSemiring (l.Germ R)
Equations
source
instance Filter.Germ.instNonAssocSemiring {α : Type u_1} {l : Filter α} {R : Type u_5} [NonAssocSemiring R] :
NonAssocSemiring (l.Germ R)
Equations
source
instance Filter.Germ.instNonUnitalNonAssocRing {α : Type u_1} {l : Filter α} {R : Type u_5} [NonUnitalNonAssocRing R] :
NonUnitalNonAssocRing (l.Germ R)
Equations
source
instance Filter.Germ.instNonUnitalRing {α : Type u_1} {l : Filter α} {R : Type u_5} [NonUnitalRing R] :
NonUnitalRing (l.Germ R)
Equations
source
instance Filter.Germ.instNonAssocRing {α : Type u_1} {l : Filter α} {R : Type u_5} [NonAssocRing R] :
NonAssocRing (l.Germ R)
Equations
source
instance Filter.Germ.instSemiring {α : Type u_1} {l : Filter α} {R : Type u_5} [Semiring R] :
Semiring (l.Germ R)
Equations
  • Filter.Germ.instSemiring = Semiring.mk ⋯ ⋯ ⋯ ⋯ Monoid.npow ⋯ ⋯
source
instance Filter.Germ.instRing {α : Type u_1} {l : Filter α} {R : Type u_5} [Ring R] :
Ring (l.Germ R)
Equations
  • Filter.Germ.instRing = Ring.mk ⋯ SubNegMonoid.zsmul ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
source
instance Filter.Germ.instNonUnitalCommSemiring {α : Type u_1} {l : Filter α} {R : Type u_5} [NonUnitalCommSemiring R] :
NonUnitalCommSemiring (l.Germ R)
Equations
source
instance Filter.Germ.instCommSemiring {α : Type u_1} {l : Filter α} {R : Type u_5} [CommSemiring R] :
CommSemiring (l.Germ R)
Equations
source
instance Filter.Germ.instNonUnitalCommRing {α : Type u_1} {l : Filter α} {R : Type u_5} [NonUnitalCommRing R] :
NonUnitalCommRing (l.Germ R)
Equations
source
instance Filter.Germ.instCommRing {α : Type u_1} {l : Filter α} {R : Type u_5} [CommRing R] :
CommRing (l.Germ R)
Equations
source
def Filter.Germ.coeRingHom {α : Type u_1} {R : Type u_5} [Semiring R] (l : Filter α) :
(α → R) →+* l.Germ R

Coercion (α → R) → Germ l R as a RingHom.

Equations
  • Filter.Germ.coeRingHom l = { toFun := Filter.Germ.ofFun, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }
source
@[simp]
theorem Filter.Germ.coe_coeRingHom {α : Type u_1} {l : Filter α} {R : Type u_5} [Semiring R] :
⇑(Filter.Germ.coeRingHom l) = Filter.Germ.ofFun
source
instance Filter.Germ.instVAdd' {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [VAdd M β] :
VAdd (l.Germ M) (l.Germ β)
Equations
source
instance Filter.Germ.instSMul' {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [SMul M β] :
SMul (l.Germ M) (l.Germ β)
Equations
source
@[simp]
theorem Filter.Germ.coe_vadd' {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [VAdd M β] (c : α → M) (f : α → β) :
↑(c +ᵥ f) = ↑c +ᵥ ↑f
source
@[simp]
theorem Filter.Germ.coe_smul' {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [SMul M β] (c : α → M) (f : α → β) :
↑(c • f) = ↑c • ↑f
source
theorem Filter.Germ.instAddAction.proof_2 {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_3} [AddMonoid M] [AddAction M β] (c₁ : M) (c₂ : M) (f : l.Germ β) :
c₁ + c₂ +ᵥ f = c₁ +ᵥ (c₂ +ᵥ f)
source
instance Filter.Germ.instAddAction {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [AddMonoid M] [AddAction M β] :
AddAction M (l.Germ β)
Equations
source
theorem Filter.Germ.instAddAction.proof_1 {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_3} [AddMonoid M] [AddAction M β] (f : l.Germ β) :
0 +ᵥ f = f
source
instance Filter.Germ.instMulAction {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [Monoid M] [MulAction M β] :
MulAction M (l.Germ β)
Equations
source
theorem Filter.Germ.instAddAction'.proof_2 {α : Type u_1} {β : Type u_3} {l : Filter α} {M : Type u_2} [AddMonoid M] [AddAction M β] (c₁ : l.Germ M) (c₂ : l.Germ M) (f : l.Germ β) :
c₁ + c₂ +ᵥ f = c₁ +ᵥ (c₂ +ᵥ f)
source
instance Filter.Germ.instAddAction' {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [AddMonoid M] [AddAction M β] :
AddAction (l.Germ M) (l.Germ β)
Equations
source
theorem Filter.Germ.instAddAction'.proof_1 {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_3} [AddMonoid M] [AddAction M β] (f : l.Germ β) :
0 +ᵥ f = f
source
instance Filter.Germ.instMulAction' {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [Monoid M] [MulAction M β] :
MulAction (l.Germ M) (l.Germ β)
Equations
source
instance Filter.Germ.instDistribMulAction {α : Type u_1} {l : Filter α} {M : Type u_5} {N : Type u_6} [Monoid M] [AddMonoid N] [DistribMulAction M N] :
DistribMulAction M (l.Germ N)
Equations
source
instance Filter.Germ.instDistribMulAction' {α : Type u_1} {l : Filter α} {M : Type u_5} {N : Type u_6} [Monoid M] [AddMonoid N] [DistribMulAction M N] :
DistribMulAction (l.Germ M) (l.Germ N)
Equations
source
instance Filter.Germ.instModule {α : Type u_1} {l : Filter α} {M : Type u_5} {R : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] :
Module R (l.Germ M)
Equations
source
instance Filter.Germ.instModule' {α : Type u_1} {l : Filter α} {M : Type u_5} {R : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] :
Module (l.Germ R) (l.Germ M)
Equations
source
instance Filter.Germ.instLE {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] :
LE (l.Germ β)
Equations
source
theorem Filter.Germ.le_def {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] :
(fun (x1 x2 : l.Germ β) => x1 ≤ x2) = Filter.Germ.LiftRel fun (x1 x2 : β) => x1 ≤ x2
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@[simp]
theorem Filter.Germ.coe_le {α : Type u_1} {β : Type u_2} {l : Filter α} {f : α → β} {g : α → β} [LE β] :
↑f ≤ ↑g ↔ f ≤ᶠ[l] g
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theorem Filter.Germ.coe_nonneg {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] [Zero β] {f : α → β} :
0 ≤ ↑f ↔ ∀ᶠ (x : α) in l, 0 ≤ f x
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theorem Filter.Germ.const_le {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] {x : β} {y : β} :
x ≤ y → ↑x ≤ ↑y
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@[simp]
theorem Filter.Germ.const_le_iff {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] [l.NeBot] {x : β} {y : β} :
↑x ≤ ↑y ↔ x ≤ y
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instance Filter.Germ.instPreorder {α : Type u_1} {β : Type u_2} {l : Filter α} [Preorder β] :
Preorder (l.Germ β)
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instance Filter.Germ.instPartialOrder {α : Type u_1} {β : Type u_2} {l : Filter α} [PartialOrder β] :
PartialOrder (l.Germ β)
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instance Filter.Germ.instBot {α : Type u_1} {β : Type u_2} {l : Filter α} [Bot β] :
Bot (l.Germ β)
Equations
  • Filter.Germ.instBot = { bot := ↑⊥ }
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instance Filter.Germ.instTop {α : Type u_1} {β : Type u_2} {l : Filter α} [Top β] :
Top (l.Germ β)
Equations
  • Filter.Germ.instTop = { top := ↑⊤ }
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@[simp]
theorem Filter.Germ.const_bot {α : Type u_1} {β : Type u_2} {l : Filter α} [Bot β] :
↑⊥ = ⊥
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@[simp]
theorem Filter.Germ.const_top {α : Type u_1} {β : Type u_2} {l : Filter α} [Top β] :
↑⊤ = ⊤
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instance Filter.Germ.instOrderBot {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] [OrderBot β] :
OrderBot (l.Germ β)
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instance Filter.Germ.instOrderTop {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] [OrderTop β] :
OrderTop (l.Germ β)
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instance Filter.Germ.instBoundedOrder {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] [BoundedOrder β] :
BoundedOrder (l.Germ β)
Equations
  • Filter.Germ.instBoundedOrder = BoundedOrder.mk
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instance Filter.Germ.instSup {α : Type u_1} {β : Type u_2} {l : Filter α} [Sup β] :
Sup (l.Germ β)
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instance Filter.Germ.instInf {α : Type u_1} {β : Type u_2} {l : Filter α} [Inf β] :
Inf (l.Germ β)
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@[simp]
theorem Filter.Germ.const_sup {α : Type u_1} {β : Type u_2} {l : Filter α} [Sup β] (a : β) (b : β) :
↑(a ⊔ b) = ↑a ⊔ ↑b
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@[simp]
theorem Filter.Germ.const_inf {α : Type u_1} {β : Type u_2} {l : Filter α} [Inf β] (a : β) (b : β) :
↑(a ⊓ b) = ↑a ⊓ ↑b
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instance Filter.Germ.instSemilatticeSup {α : Type u_1} {β : Type u_2} {l : Filter α} [SemilatticeSup β] :
SemilatticeSup (l.Germ β)
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instance Filter.Germ.instSemilatticeInf {α : Type u_1} {β : Type u_2} {l : Filter α} [SemilatticeInf β] :
SemilatticeInf (l.Germ β)
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instance Filter.Germ.instLattice {α : Type u_1} {β : Type u_2} {l : Filter α} [Lattice β] :
Lattice (l.Germ β)
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instance Filter.Germ.instDistribLattice {α : Type u_1} {β : Type u_2} {l : Filter α} [DistribLattice β] :
DistribLattice (l.Germ β)
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instance Filter.Germ.instExistsAddOfLE {α : Type u_1} {β : Type u_2} {l : Filter α} [Add β] [LE β] [ExistsAddOfLE β] :
ExistsAddOfLE (l.Germ β)
Equations
  • ⋯ = ⋯
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instance Filter.Germ.instExistsMulOfLE {α : Type u_1} {β : Type u_2} {l : Filter α} [Mul β] [LE β] [ExistsMulOfLE β] :
ExistsMulOfLE (l.Germ β)
Equations
  • ⋯ = ⋯