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

• Filter.Germ l β to be the space of germs of functions α → β at a filter l : Filter α;
• coercion from α → β to Germ l β: (f : Germ l β) is the germ of f : α → β at l : Filter α; this coercion is declared as CoeTC;
• (const l c : Germ l β) is the germ of the constant function fun x : α ↦ c at a filter l;
• coercion from β to Germ l β: (↑c : Germ l β) is the germ of the constant function fun x : α ↦ c at a filter l; this coercion is declared as CoeTC;
• map (F : β → γ) (f : Germ l β) to be the composition of a function F and a germ f;
• map₂ (F : β → γ → δ) (f : Germ l β) (g : Germ l γ) to be the germ of fun x ↦ F (f x) (g x) at l;
• f.Tendsto lb: we say that a germ f : Germ l β tends to a filter lb if its representatives tend to lb along l;
• f.compTendsto g hg and f.compTendsto' g hg: given f : Germ l β and a function g : γ → α (resp., a germ g : Germ lc α), if g tends to l along lc, then the composition f ∘ g is a well-defined germ at lc;
• Germ.liftPred, Germ.liftRel: lift a predicate or a relation to the space of germs: (f : Germ l β).liftPred p means ∀ᶠ x in l, p (f x), and similarly for a relation.

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:

• one-operation algebraic structures up to CommGroup;
• MulZeroClass, Distrib, Semiring, CommSemiring, Ring, CommRing;
• MulAction, DistribMulAction, Module;
• Preorder, PartialOrder, and Lattice structures, as well as BoundedOrder;
• OrderedCancelCommMonoid and OrderedCancelAddCommMonoid.

Tags

filter, germ

theorem Filter.const_eventuallyEq' {α : Type u_1} {β : Type u_2} {l : Filter α} [Filter.NeBot l] {a : β} {b : β} : (∀ᶠ (x : α) in l, a = b) ↔ a = b

theorem Filter.const_eventuallyEq {α : Type u_1} {β : Type u_2} {l : Filter α} [Filter.NeBot l] {a : β} {b : β} : ((fun (x : α) => a) =ᶠ[l] fun (x : α) => b) ↔ a = b

theorem Filter.EventuallyEq.comp_tendsto {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} {f : α → β} {f' : α → β} (H : f =ᶠ[l] f') {g : γ → α} {lc : Filter γ} (hg : Filter.Tendsto g lc l) : f ∘ g =ᶠ[lc] f' ∘ g

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

• Filter.germSetoid l β = { r := Filter.EventuallyEq l, iseqv := (_ : Equivalence (Filter.EventuallyEq l)) }

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

• Filter.Germ l β = Quotient (Filter.germSetoid l β)

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

• Filter.productSetoid l ε = { r := fun (f g : (a : α) → ε a) => ∀ᶠ (a : α) in l, f a = g a, iseqv := (_ : Equivalence fun (f g : (a : α) → ε a) => ∀ᶠ (a : α) in l, f a = g a) }

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

• Filter.Product l ε = Quotient (Filter.productSetoid l ε)

instance Filter.Product.coeTC {α : Type u_1} {l : Filter α} {ε : α → Type u_5} : CoeTC ((a : α) → ε a) (Filter.Product l ε)

Equations

• Filter.Product.coeTC = { coe := Quotient.mk' }

instance Filter.Product.inhabited {α : Type u_1} {l : Filter α} {ε : α → Type u_5} [(a : α) → Inhabited (ε a)] : Inhabited (Filter.Product l ε)

Equations

• Filter.Product.inhabited = { default := Quotient.mk' fun (a : α) => default }

def Filter.Germ.ofFun {α : Type u_1} {β : Type u_2} {l : Filter α} : (α → β) → Filter.Germ l β

Equations

• Filter.Germ.ofFun = Quotient.mk'

instance Filter.Germ.instCoeTCForAllGerm {α : Type u_1} {β : Type u_2} {l : Filter α} : CoeTC (α → β) (Filter.Germ l β)

Equations

• Filter.Germ.instCoeTCForAllGerm = { coe := Filter.Germ.ofFun }

def Filter.Germ.const {α : Type u_1} {β : Type u_2} {l : Filter α} (b : β) : Filter.Germ l β

Equations

• ↑b = ↑fun (x : α) => b

instance Filter.Germ.coeTC {α : Type u_1} {β : Type u_2} {l : Filter α} : CoeTC β (Filter.Germ l β)

Equations

• Filter.Germ.coeTC = { coe := Filter.Germ.const }

@[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

@[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

theorem Filter.Germ.inductionOn {α : Type u_1} {β : Type u_2} {l : Filter α} (f : Filter.Germ l β) {p : Filter.Germ l β → Prop} (h : ∀ (f : α → β), p ↑f) : p f

theorem Filter.Germ.inductionOn₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (f : Filter.Germ l β) (g : Filter.Germ l γ) {p : Filter.Germ l β → Filter.Germ l γ → Prop} (h : ∀ (f : α → β) (g : α → γ), p ↑f ↑g) : p f g

theorem Filter.Germ.inductionOn₃ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {l : Filter α} (f : Filter.Germ l β) (g : Filter.Germ l γ) (h : Filter.Germ l δ) {p : Filter.Germ l β → Filter.Germ l γ → Filter.Germ l δ → Prop} (H : ∀ (f : α → β) (g : α → γ) (h : α → δ), p ↑f ↑g ↑h) : p f g h

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 : (Filter.EventuallyEq l ⇒ Filter.EventuallyEq lc) F F) : Filter.Germ l β → Filter.Germ lc 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

• Filter.Germ.map' F hF = Quotient.map' F hF

def Filter.Germ.liftOn {α : Type u_1} {β : Type u_2} {l : Filter α} {γ : Sort u_5} (f : Filter.Germ l β) (F : (α → β) → γ) (hF : (Filter.EventuallyEq l ⇒ fun (x x_1 : γ) => x = x_1) 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

• Filter.Germ.liftOn f F hF = Quotient.liftOn' f F hF

@[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 : (Filter.EventuallyEq l ⇒ Filter.EventuallyEq lc) F F) (f : α → β) : Filter.Germ.map' F hF ↑f = ↑(F f)

@[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

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.

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

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

Equations

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

@[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)

@[simp]

theorem Filter.Germ.map_id {α : Type u_1} {β : Type u_2} {l : Filter α} : Filter.Germ.map id = id

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 : Filter.Germ l β) : Filter.Germ.map op₁ (Filter.Germ.map op₂ f) = Filter.Germ.map (op₁ ∘ op₂) f

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) : Filter.Germ l β → Filter.Germ l γ → Filter.Germ l δ

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

Equations

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

@[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)

def Filter.Germ.Tendsto {α : Type u_1} {β : Type u_2} {l : Filter α} (f : Filter.Germ l 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

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

@[simp]

theorem Filter.Germ.coe_tendsto {α : Type u_1} {β : Type u_2} {l : Filter α} {f : α → β} {lb : Filter β} : Filter.Germ.Tendsto (↑f) lb ↔ Filter.Tendsto f l lb

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 → Filter.Germ.Tendsto (↑f) lb

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

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

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

@[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 : Filter.Germ lc α} (hg : Filter.Germ.Tendsto g l) : Filter.Germ.compTendsto' (↑f) g hg = Filter.Germ.map f g

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

• Filter.Germ.compTendsto f g hg = Filter.Germ.compTendsto' f ↑g (_ : Filter.Germ.Tendsto (↑g) l)

@[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) : Filter.Germ.compTendsto (↑f) g hg = ↑(f ∘ g)

@[simp]

theorem Filter.Germ.compTendsto'_coe {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (f : Filter.Germ l β) {lc : Filter γ} {g : γ → α} (hg : Filter.Tendsto g lc l) : Filter.Germ.compTendsto' f ↑g (_ : Filter.Germ.Tendsto (↑g) l) = Filter.Germ.compTendsto f g hg

@[simp]

theorem Filter.Germ.const_inj {α : Type u_1} {β : Type u_2} {l : Filter α} [Filter.NeBot l] {a : β} {b : β} : ↑a = ↑b ↔ a = b

@[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)

@[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)

@[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) : Filter.Germ.compTendsto (↑b) g hg = ↑b

@[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 : Filter.Germ lc α} (hg : Filter.Germ.Tendsto g l) : Filter.Germ.compTendsto' (↑b) g hg = ↑b

def Filter.Germ.LiftPred {α : Type u_1} {β : Type u_2} {l : Filter α} (p : β → Prop) (f : Filter.Germ l β) : Prop

Lift a predicate on β to Germ l β.

Equations

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

@[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)

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

@[simp]

theorem Filter.Germ.liftPred_const_iff {α : Type u_1} {β : Type u_2} {l : Filter α} [Filter.NeBot l] {p : β → Prop} {x : β} : Filter.Germ.LiftPred p ↑x ↔ p x

def Filter.Germ.LiftRel {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} (r : β → γ → Prop) (f : Filter.Germ l β) (g : Filter.Germ l γ) : Prop

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

Equations

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

@[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)

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

@[simp]

theorem Filter.Germ.liftRel_const_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} {l : Filter α} [Filter.NeBot l] {r : β → γ → Prop} {x : β} {y : γ} : Filter.Germ.LiftRel r ↑x ↑y ↔ r x y

instance Filter.Germ.inhabited {α : Type u_1} {β : Type u_2} {l : Filter α} [Inhabited β] : Inhabited (Filter.Germ l β)

Equations

• Filter.Germ.inhabited = { default := ↑default }

instance Filter.Germ.add {α : Type u_1} {l : Filter α} {M : Type u_5} [Add M] : Add (Filter.Germ l M)

Equations

• Filter.Germ.add = { add := Filter.Germ.map₂ fun (x x_1 : M) => x + x_1 }

instance Filter.Germ.mul {α : Type u_1} {l : Filter α} {M : Type u_5} [Mul M] : Mul (Filter.Germ l M)

Equations

• Filter.Germ.mul = { mul := Filter.Germ.map₂ fun (x x_1 : M) => x * x_1 }

@[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

@[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

instance Filter.Germ.zero {α : Type u_1} {l : Filter α} {M : Type u_5} [Zero M] : Zero (Filter.Germ l M)

Equations

• Filter.Germ.zero = { zero := ↑0 }

instance Filter.Germ.one {α : Type u_1} {l : Filter α} {M : Type u_5} [One M] : One (Filter.Germ l M)

Equations

• Filter.Germ.one = { one := ↑1 }

@[simp]

theorem Filter.Germ.coe_zero {α : Type u_1} {l : Filter α} {M : Type u_5} [Zero M] : ↑0 = 0

@[simp]

theorem Filter.Germ.coe_one {α : Type u_1} {l : Filter α} {M : Type u_5} [One M] : ↑1 = 1

instance Filter.Germ.addSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [AddSemigroup M] : AddSemigroup (Filter.Germ l M)

Equations

• Filter.Germ.addSemigroup = AddSemigroup.mk (_ : ∀ (a b c : Filter.Germ l M), a + b + c = a + (b + c))

theorem Filter.Germ.addSemigroup.proof_1 {α : Type u_2} {l : Filter α} {M : Type u_1} [AddSemigroup M] (a : Filter.Germ l M) (b : Filter.Germ l M) (c : Filter.Germ l M) : a + b + c = a + (b + c)

instance Filter.Germ.semigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [Semigroup M] : Semigroup (Filter.Germ l M)

Equations

• Filter.Germ.semigroup = Semigroup.mk (_ : ∀ (a b c : Filter.Germ l M), a * b * c = a * (b * c))

theorem Filter.Germ.addCommSemigroup.proof_1 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddCommSemigroup M] (q₁ : Quotient (Filter.germSetoid l M)) (q₂ : Quotient (Filter.germSetoid l M)) : q₁ + q₂ = q₂ + q₁

instance Filter.Germ.addCommSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [AddCommSemigroup M] : AddCommSemigroup (Filter.Germ l M)

Equations

• Filter.Germ.addCommSemigroup = AddCommSemigroup.mk (_ : ∀ (q₁ q₂ : Quotient (Filter.germSetoid l M)), q₁ + q₂ = q₂ + q₁)

instance Filter.Germ.commSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [CommSemigroup M] : CommSemigroup (Filter.Germ l M)

Equations

• Filter.Germ.commSemigroup = CommSemigroup.mk (_ : ∀ (q₁ q₂ : Quotient (Filter.germSetoid l M)), q₁ * q₂ = q₂ * q₁)

instance Filter.Germ.instIsAddLeftCancel {α : Type u_1} {l : Filter α} {M : Type u_5} [Add M] [IsLeftCancelAdd M] : IsLeftCancelAdd (Filter.Germ l M)

Equations

• (_ : IsLeftCancelAdd (Filter.Germ l M)) = (_ : IsLeftCancelAdd (Filter.Germ l M))

instance Filter.Germ.instIsLeftCancelMul {α : Type u_1} {l : Filter α} {M : Type u_5} [Mul M] [IsLeftCancelMul M] : IsLeftCancelMul (Filter.Germ l M)

Equations

• (_ : IsLeftCancelMul (Filter.Germ l M)) = (_ : IsLeftCancelMul (Filter.Germ l M))

instance Filter.Germ.instIsAddRightCancel {α : Type u_1} {l : Filter α} {M : Type u_5} [Add M] [IsRightCancelAdd M] : IsRightCancelAdd (Filter.Germ l M)

Equations

• (_ : IsRightCancelAdd (Filter.Germ l M)) = (_ : IsRightCancelAdd (Filter.Germ l M))

instance Filter.Germ.instIsRightCancelMul {α : Type u_1} {l : Filter α} {M : Type u_5} [Mul M] [IsRightCancelMul M] : IsRightCancelMul (Filter.Germ l M)

Equations

• (_ : IsRightCancelMul (Filter.Germ l M)) = (_ : IsRightCancelMul (Filter.Germ l M))

instance Filter.Germ.instIsAddCancel {α : Type u_1} {l : Filter α} {M : Type u_5} [Add M] [IsCancelAdd M] : IsCancelAdd (Filter.Germ l M)

Equations

• (_ : IsCancelAdd (Filter.Germ l M)) = (_ : IsCancelAdd (Filter.Germ l M))

instance Filter.Germ.instIsCancelMul {α : Type u_1} {l : Filter α} {M : Type u_5} [Mul M] [IsCancelMul M] : IsCancelMul (Filter.Germ l M)

Equations

• (_ : IsCancelMul (Filter.Germ l M)) = (_ : IsCancelMul (Filter.Germ l M))

instance Filter.Germ.addLeftCancelSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [AddLeftCancelSemigroup M] : AddLeftCancelSemigroup (Filter.Germ l M)

Equations

• Filter.Germ.addLeftCancelSemigroup = let src := Filter.Germ.addSemigroup; AddLeftCancelSemigroup.mk (_ : ∀ (x x_1 x_2 : Filter.Germ l M), x + x_1 = x + x_2 → x_1 = x_2)

theorem Filter.Germ.addLeftCancelSemigroup.proof_1 {α : Type u_2} {l : Filter α} {M : Type u_1} [AddLeftCancelSemigroup M] : ∀ (x x_1 x_2 : Filter.Germ l M), x + x_1 = x + x_2 → x_1 = x_2

instance Filter.Germ.leftCancelSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [LeftCancelSemigroup M] : LeftCancelSemigroup (Filter.Germ l M)

Equations

• Filter.Germ.leftCancelSemigroup = let src := Filter.Germ.semigroup; LeftCancelSemigroup.mk (_ : ∀ (x x_1 x_2 : Filter.Germ l M), x * x_1 = x * x_2 → x_1 = x_2)

theorem Filter.Germ.addRightCancelSemigroup.proof_1 {α : Type u_2} {l : Filter α} {M : Type u_1} [AddRightCancelSemigroup M] : ∀ (x x_1 x_2 : Filter.Germ l M), x + x_1 = x_2 + x_1 → x = x_2

instance Filter.Germ.addRightCancelSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [AddRightCancelSemigroup M] : AddRightCancelSemigroup (Filter.Germ l M)

Equations

• Filter.Germ.addRightCancelSemigroup = let src := Filter.Germ.addSemigroup; AddRightCancelSemigroup.mk (_ : ∀ (x x_1 x_2 : Filter.Germ l M), x + x_1 = x_2 + x_1 → x = x_2)

instance Filter.Germ.rightCancelSemigroup {α : Type u_1} {l : Filter α} {M : Type u_5} [RightCancelSemigroup M] : RightCancelSemigroup (Filter.Germ l M)

Equations

• Filter.Germ.rightCancelSemigroup = let src := Filter.Germ.semigroup; RightCancelSemigroup.mk (_ : ∀ (x x_1 x_2 : Filter.Germ l M), x * x_1 = x_2 * x_1 → x = x_2)

instance Filter.Germ.addZeroClass {α : Type u_1} {l : Filter α} {M : Type u_5} [AddZeroClass M] : AddZeroClass (Filter.Germ l M)

Equations

• Filter.Germ.addZeroClass = AddZeroClass.mk (_ : ∀ (q : Quotient (Filter.germSetoid l M)), 0 + q = q) (_ : ∀ (q : Quotient (Filter.germSetoid l M)), q + 0 = q)

theorem Filter.Germ.addZeroClass.proof_1 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddZeroClass M] (q : Quotient (Filter.germSetoid l M)) : 0 + q = q

theorem Filter.Germ.addZeroClass.proof_2 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddZeroClass M] (q : Quotient (Filter.germSetoid l M)) : q + 0 = q

instance Filter.Germ.mulOneClass {α : Type u_1} {l : Filter α} {M : Type u_5} [MulOneClass M] : MulOneClass (Filter.Germ l M)

Equations

• Filter.Germ.mulOneClass = MulOneClass.mk (_ : ∀ (q : Quotient (Filter.germSetoid l M)), 1 * q = q) (_ : ∀ (q : Quotient (Filter.germSetoid l M)), q * 1 = q)

instance Filter.Germ.vadd {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [VAdd M G] : VAdd M (Filter.Germ l G)

Equations

• Filter.Germ.vadd = { vadd := fun (n : M) => Filter.Germ.map fun (x : G) => n +ᵥ x }

instance Filter.Germ.smul {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [SMul M G] : SMul M (Filter.Germ l G)

Equations

• Filter.Germ.smul = { smul := fun (n : M) => Filter.Germ.map fun (x : G) => n • x }

instance Filter.Germ.pow {α : Type u_1} {l : Filter α} {M : Type u_5} {G : Type u_6} [Pow G M] : Pow (Filter.Germ l G) M

Equations

• Filter.Germ.pow = { pow := fun (f : Filter.Germ l G) (n : M) => Filter.Germ.map (fun (x : G) => x ^ n) f }

@[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

@[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

@[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

@[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

@[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

@[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

@[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

@[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

theorem Filter.Germ.addMonoid.proof_4 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddMonoid M] : ∀ (x : α → M) (x_1 : ℕ), ↑(x_1 • x) = ↑(x_1 • x)

instance Filter.Germ.addMonoid {α : Type u_1} {l : Filter α} {M : Type u_5} [AddMonoid M] : AddMonoid (Filter.Germ l M)

Equations

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

theorem Filter.Germ.addMonoid.proof_7 {α : Type u_2} {l : Filter α} {M : Type u_1} [AddMonoid M] (x : Filter.Germ l M) : AddMonoid.nsmul 0 x = 0

theorem Filter.Germ.addMonoid.proof_8 {α : Type u_2} {l : Filter α} {M : Type u_1} [AddMonoid M] (n : ℕ) (x : Filter.Germ l M) : AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x

theorem Filter.Germ.addMonoid.proof_2 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddMonoid M] : ↑0 = ↑0

theorem Filter.Germ.addMonoid.proof_5 {α : Type u_2} {l : Filter α} {M : Type u_1} [AddMonoid M] (a : Filter.Germ l M) : 0 + a = a

theorem Filter.Germ.addMonoid.proof_1 {α : Type u_1} {l : Filter α} {M : Type u_2} : Function.Surjective (Quot.mk Setoid.r)

theorem Filter.Germ.addMonoid.proof_3 {α : Type u_1} {l : Filter α} {M : Type u_2} [AddMonoid M] : ∀ (x x_1 : α → M), ↑(x + x_1) = ↑(x + x_1)

theorem Filter.Germ.addMonoid.proof_6 {α : Type u_2} {l : Filter α} {M : Type u_1} [AddMonoid M] (a : Filter.Germ l M) : a + 0 = a

instance Filter.Germ.monoid {α : Type u_1} {l : Filter α} {M : Type u_5} [Monoid M] : Monoid (Filter.Germ l M)

Equations

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

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' := (_ : ↑0 = ↑0) }.toFun (x + x_1) = { toFun := Filter.Germ.ofFun, map_zero' := (_ : ↑0 = ↑0) }.toFun (x + x_1)

theorem Filter.Germ.coeAddHom.proof_1 {α : Type u_1} {M : Type u_2} [AddMonoid M] (l : Filter α) : ↑0 = ↑0

def Filter.Germ.coeAddHom {α : Type u_1} {M : Type u_5} [AddMonoid M] (l : Filter α) : (α → M) →+ Filter.Germ l M

Coercion from functions to germs as an additive monoid homomorphism.

Equations

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

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

Coercion from functions to germs as a monoid homomorphism.

Equations

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

@[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

@[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

instance Filter.Germ.addCommMonoid {α : Type u_1} {l : Filter α} {M : Type u_5} [AddCommMonoid M] : AddCommMonoid (Filter.Germ l M)

Equations

• Filter.Germ.addCommMonoid = AddCommMonoid.mk (_ : ∀ (a b : Filter.Germ l M), a + b = b + a)

theorem Filter.Germ.addCommMonoid.proof_1 {α : Type u_2} {l : Filter α} {M : Type u_1} [AddCommMonoid M] (a : Filter.Germ l M) (b : Filter.Germ l M) : a + b = b + a

instance Filter.Germ.commMonoid {α : Type u_1} {l : Filter α} {M : Type u_5} [CommMonoid M] : CommMonoid (Filter.Germ l M)

Equations

• Filter.Germ.commMonoid = CommMonoid.mk (_ : ∀ (a b : Filter.Germ l M), a * b = b * a)

instance Filter.Germ.natCast {α : Type u_1} {l : Filter α} {M : Type u_5} [NatCast M] : NatCast (Filter.Germ l M)

Equations

• Filter.Germ.natCast = { natCast := fun (n : ℕ) => ↑↑n }

@[simp]

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

@[simp]

theorem Filter.Germ.const_nat {α : Type u_1} {l : Filter α} {M : Type u_5} [NatCast M] (n : ℕ) : ↑↑n = ↑n

@[simp]

theorem Filter.Germ.coe_ofNat {α : Type u_1} {l : Filter α} {M : Type u_5} [NatCast M] (n : ℕ) [Nat.AtLeastTwo n] : ↑(OfNat.ofNat n) = OfNat.ofNat n

@[simp]

theorem Filter.Germ.const_ofNat {α : Type u_1} {l : Filter α} {M : Type u_5} [NatCast M] (n : ℕ) [Nat.AtLeastTwo n] : ↑(OfNat.ofNat n) = OfNat.ofNat n

instance Filter.Germ.intCast {α : Type u_1} {l : Filter α} {M : Type u_5} [IntCast M] : IntCast (Filter.Germ l M)

Equations

• Filter.Germ.intCast = { intCast := fun (n : ℤ) => ↑↑n }

@[simp]

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

instance Filter.Germ.addMonoidWithOne {α : Type u_1} {l : Filter α} {M : Type u_5} [AddMonoidWithOne M] : AddMonoidWithOne (Filter.Germ l M)

Equations

• Filter.Germ.addMonoidWithOne = let src := Filter.Germ.natCast; let src_1 := Filter.Germ.addMonoid; let src_2 := Filter.Germ.one; AddMonoidWithOne.mk

instance Filter.Germ.addCommMonoidWithOne {α : Type u_1} {l : Filter α} {M : Type u_5} [AddCommMonoidWithOne M] : AddCommMonoidWithOne (Filter.Germ l M)

Equations

• Filter.Germ.addCommMonoidWithOne = AddCommMonoidWithOne.mk (_ : ∀ (a b : Filter.Germ l M), a + b = b + a)

instance Filter.Germ.neg {α : Type u_1} {l : Filter α} {G : Type u_6} [Neg G] : Neg (Filter.Germ l G)

Equations

• Filter.Germ.neg = { neg := Filter.Germ.map Neg.neg }

instance Filter.Germ.inv {α : Type u_1} {l : Filter α} {G : Type u_6} [Inv G] : Inv (Filter.Germ l G)

Equations

• Filter.Germ.inv = { inv := Filter.Germ.map Inv.inv }

@[simp]

theorem Filter.Germ.coe_neg {α : Type u_1} {l : Filter α} {G : Type u_6} [Neg G] (f : α → G) : ↑(-f) = -↑f

@[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

@[simp]

theorem Filter.Germ.const_neg {α : Type u_1} {l : Filter α} {G : Type u_6} [Neg G] (a : G) : ↑(-a) = -↑a

@[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

instance Filter.Germ.sub {α : Type u_1} {l : Filter α} {M : Type u_5} [Sub M] : Sub (Filter.Germ l M)

Equations

• Filter.Germ.sub = { sub := Filter.Germ.map₂ fun (x x_1 : M) => x - x_1 }

instance Filter.Germ.div {α : Type u_1} {l : Filter α} {M : Type u_5} [Div M] : Div (Filter.Germ l M)

Equations

• Filter.Germ.div = { div := Filter.Germ.map₂ fun (x x_1 : M) => x / x_1 }

@[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

@[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

@[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

@[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

theorem Filter.Germ.involutiveNeg.proof_1 {α : Type u_1} {l : Filter α} {G : Type u_2} [InvolutiveNeg G] (q : Quotient (Filter.germSetoid l G)) : - -q = q

instance Filter.Germ.involutiveNeg {α : Type u_1} {l : Filter α} {G : Type u_6} [InvolutiveNeg G] : InvolutiveNeg (Filter.Germ l G)

Equations

• Filter.Germ.involutiveNeg = InvolutiveNeg.mk (_ : ∀ (q : Quotient (Filter.germSetoid l G)), - -q = q)

instance Filter.Germ.involutiveInv {α : Type u_1} {l : Filter α} {G : Type u_6} [InvolutiveInv G] : InvolutiveInv (Filter.Germ l G)

Equations

• Filter.Germ.involutiveInv = InvolutiveInv.mk (_ : ∀ (q : Quotient (Filter.germSetoid l G)), q⁻¹⁻¹ = q)

instance Filter.Germ.hasDistribNeg {α : Type u_1} {l : Filter α} {G : Type u_6} [Mul G] [HasDistribNeg G] : HasDistribNeg (Filter.Germ l G)

Equations

• Filter.Germ.hasDistribNeg = HasDistribNeg.mk (_ : ∀ (q₁ q₂ : Quotient (Filter.germSetoid l G)), -q₁ * q₂ = -(q₁ * q₂)) (_ : ∀ (q₁ q₂ : Quotient (Filter.germSetoid l G)), q₁ * -q₂ = -(q₁ * q₂))

instance Filter.Germ.negZeroClass {α : Type u_1} {l : Filter α} {G : Type u_6} [NegZeroClass G] : NegZeroClass (Filter.Germ l G)

Equations

• Filter.Germ.negZeroClass = NegZeroClass.mk (_ : ↑((fun (x : α → G) => Neg.neg ∘ x) fun (x : α) => 0) = ↑fun (x : α) => 0)

theorem Filter.Germ.negZeroClass.proof_1 {α : Type u_1} {l : Filter α} {G : Type u_2} [NegZeroClass G] : ↑((fun (x : α → G) => Neg.neg ∘ x) fun (x : α) => 0) = ↑fun (x : α) => 0

instance Filter.Germ.invOneClass {α : Type u_1} {l : Filter α} {G : Type u_6} [InvOneClass G] : InvOneClass (Filter.Germ l G)

Equations

• Filter.Germ.invOneClass = InvOneClass.mk (_ : ↑((fun (x : α → G) => Inv.inv ∘ x) fun (x : α) => 1) = ↑fun (x : α) => 1)

theorem Filter.Germ.subNegMonoid.proof_1 {α : Type u_1} {l : Filter α} {G : Type u_2} [SubNegMonoid G] (q₁ : Quotient (Filter.germSetoid l G)) (q₂ : Quotient (Filter.germSetoid l G)) : q₁ - q₂ = q₁ + -q₂

theorem Filter.Germ.subNegMonoid.proof_3 {α : Type u_1} {l : Filter α} {G : Type u_2} [SubNegMonoid G] : ∀ (x : ℕ) (q : Quotient (Filter.germSetoid l G)), (fun (z : ℤ) (f : Filter.Germ l G) => z • f) (Int.ofNat (Nat.succ x)) q = q + (fun (z : ℤ) (f : Filter.Germ l G) => z • f) (Int.ofNat x) q

theorem Filter.Germ.subNegMonoid.proof_4 {α : Type u_1} {l : Filter α} {G : Type u_2} [SubNegMonoid G] : ∀ (x : ℕ) (q : Quotient (Filter.germSetoid l G)), (fun (z : ℤ) (f : Filter.Germ l G) => z • f) (Int.negSucc x) q = -(fun (z : ℤ) (f : Filter.Germ l G) => z • f) (↑(Nat.succ x)) q

instance Filter.Germ.subNegMonoid {α : Type u_1} {l : Filter α} {G : Type u_6} [SubNegMonoid G] : SubNegMonoid (Filter.Germ l G)

Equations

• Filter.Germ.subNegMonoid = let src := Filter.Germ.addMonoid; let src_1 := Filter.Germ.neg; let src_2 := Filter.Germ.sub; SubNegMonoid.mk fun (z : ℤ) (f : Filter.Germ l G) => z • f

theorem Filter.Germ.subNegMonoid.proof_2 {α : Type u_1} {l : Filter α} {G : Type u_2} [SubNegMonoid G] (q : Quotient (Filter.germSetoid l G)) : (fun (z : ℤ) (f : Filter.Germ l G) => z • f) 0 q = 0

instance Filter.Germ.divInvMonoid {α : Type u_1} {l : Filter α} {G : Type u_6} [DivInvMonoid G] : DivInvMonoid (Filter.Germ l G)

Equations

• Filter.Germ.divInvMonoid = let src := Filter.Germ.monoid; let src_1 := Filter.Germ.inv; let src_2 := Filter.Germ.div; DivInvMonoid.mk fun (z : ℤ) (f : Filter.Germ l G) => f ^ z

theorem Filter.Germ.divisionAddMonoid.proof_1 {α : Type u_2} {l : Filter α} {G : Type u_1} [SubtractionMonoid G] (a : Filter.Germ l G) : - -a = a

instance Filter.Germ.divisionAddMonoid {α : Type u_1} {l : Filter α} {G : Type u_6} [SubtractionMonoid G] : SubtractionMonoid (Filter.Germ l G)

Equations

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

theorem Filter.Germ.divisionAddMonoid.proof_2 {α : Type u_2} {l : Filter α} {G : Type u_1} [SubtractionMonoid G] (x : Filter.Germ l G) (y : Filter.Germ l G) : -(x + y) = -y + -x

theorem Filter.Germ.divisionAddMonoid.proof_3 {α : Type u_2} {l : Filter α} {G : Type u_1} [SubtractionMonoid G] (x : Filter.Germ l G) (y : Filter.Germ l G) : x + y = 0 → -x = y

instance Filter.Germ.divisionMonoid {α : Type u_1} {l : Filter α} {G : Type u_6} [DivisionMonoid G] : DivisionMonoid (Filter.Germ l G)

Equations

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

theorem Filter.Germ.addGroup.proof_1 {α : Type u_1} {l : Filter α} {G : Type u_2} [AddGroup G] (q : Quotient (Filter.germSetoid l G)) : -q + q = 0

instance Filter.Germ.addGroup {α : Type u_1} {l : Filter α} {G : Type u_6} [AddGroup G] : AddGroup (Filter.Germ l G)

Equations

• Filter.Germ.addGroup = AddGroup.mk (_ : ∀ (q : Quotient (Filter.germSetoid l G)), -q + q = 0)

instance Filter.Germ.group {α : Type u_1} {l : Filter α} {G : Type u_6} [Group G] : Group (Filter.Germ l G)

Equations

• Filter.Germ.group = Group.mk (_ : ∀ (q : Quotient (Filter.germSetoid l G)), q⁻¹ * q = 1)

instance Filter.Germ.addCommGroup {α : Type u_1} {l : Filter α} {G : Type u_6} [AddCommGroup G] : AddCommGroup (Filter.Germ l G)

Equations

• Filter.Germ.addCommGroup = AddCommGroup.mk (_ : ∀ (a b : Filter.Germ l G), a + b = b + a)

theorem Filter.Germ.addCommGroup.proof_1 {α : Type u_2} {l : Filter α} {G : Type u_1} [AddCommGroup G] (a : Filter.Germ l G) (b : Filter.Germ l G) : a + b = b + a

instance Filter.Germ.commGroup {α : Type u_1} {l : Filter α} {G : Type u_6} [CommGroup G] : CommGroup (Filter.Germ l G)

Equations

• Filter.Germ.commGroup = CommGroup.mk (_ : ∀ (a b : Filter.Germ l G), a * b = b * a)

instance Filter.Germ.addGroupWithOne {α : Type u_1} {l : Filter α} {G : Type u_6} [AddGroupWithOne G] : AddGroupWithOne (Filter.Germ l G)

Equations

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

instance Filter.Germ.nontrivial {α : Type u_1} {l : Filter α} {R : Type u_5} [Nontrivial R] [Filter.NeBot l] : Nontrivial (Filter.Germ l R)

Equations

• (_ : Nontrivial (Filter.Germ l R)) = (_ : Nontrivial (Filter.Germ l R))

instance Filter.Germ.mulZeroClass {α : Type u_1} {l : Filter α} {R : Type u_5} [MulZeroClass R] : MulZeroClass (Filter.Germ l R)

Equations

• Filter.Germ.mulZeroClass = MulZeroClass.mk (_ : ∀ (q : Quotient (Filter.germSetoid l R)), 0 * q = 0) (_ : ∀ (q : Quotient (Filter.germSetoid l R)), q * 0 = 0)

instance Filter.Germ.mulZeroOneClass {α : Type u_1} {l : Filter α} {R : Type u_5} [MulZeroOneClass R] : MulZeroOneClass (Filter.Germ l R)

Equations

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

instance Filter.Germ.monoidWithZero {α : Type u_1} {l : Filter α} {R : Type u_5} [MonoidWithZero R] : MonoidWithZero (Filter.Germ l R)

Equations

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

instance Filter.Germ.distrib {α : Type u_1} {l : Filter α} {R : Type u_5} [Distrib R] : Distrib (Filter.Germ l R)

Equations

• Filter.Germ.distrib = Distrib.mk (_ : ∀ (a b c : Filter.Germ l R), a * (b + c) = a * b + a * c) (_ : ∀ (a b c : Filter.Germ l R), (a + b) * c = a * c + b * c)

instance Filter.Germ.nonUnitalNonAssocSemiring {α : Type u_1} {l : Filter α} {R : Type u_5} [NonUnitalNonAssocSemiring R] : NonUnitalNonAssocSemiring (Filter.Germ l R)

Equations

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

instance Filter.Germ.nonUnitalSemiring {α : Type u_1} {l : Filter α} {R : Type u_5} [NonUnitalSemiring R] : NonUnitalSemiring (Filter.Germ l R)

Equations

• Filter.Germ.nonUnitalSemiring = NonUnitalSemiring.mk (_ : ∀ (a b c : Filter.Germ l R), a * b * c = a * (b * c))

instance Filter.Germ.nonAssocSemiring {α : Type u_1} {l : Filter α} {R : Type u_5} [NonAssocSemiring R] : NonAssocSemiring (Filter.Germ l R)

Equations

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

instance Filter.Germ.nonUnitalNonAssocRing {α : Type u_1} {l : Filter α} {R : Type u_5} [NonUnitalNonAssocRing R] : NonUnitalNonAssocRing (Filter.Germ l R)

Equations

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

instance Filter.Germ.nonUnitalRing {α : Type u_1} {l : Filter α} {R : Type u_5} [NonUnitalRing R] : NonUnitalRing (Filter.Germ l R)

Equations

• Filter.Germ.nonUnitalRing = NonUnitalRing.mk (_ : ∀ (a b c : Filter.Germ l R), a * b * c = a * (b * c))

instance Filter.Germ.nonAssocRing {α : Type u_1} {l : Filter α} {R : Type u_5} [NonAssocRing R] : NonAssocRing (Filter.Germ l R)

Equations

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

instance Filter.Germ.semiring {α : Type u_1} {l : Filter α} {R : Type u_5} [Semiring R] : Semiring (Filter.Germ l R)

Equations

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

instance Filter.Germ.ring {α : Type u_1} {l : Filter α} {R : Type u_5} [Ring R] : Ring (Filter.Germ l R)

Equations

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

instance Filter.Germ.nonUnitalCommSemiring {α : Type u_1} {l : Filter α} {R : Type u_5} [NonUnitalCommSemiring R] : NonUnitalCommSemiring (Filter.Germ l R)

Equations

• Filter.Germ.nonUnitalCommSemiring = NonUnitalCommSemiring.mk (_ : ∀ (a b : Filter.Germ l R), a * b = b * a)

instance Filter.Germ.commSemiring {α : Type u_1} {l : Filter α} {R : Type u_5} [CommSemiring R] : CommSemiring (Filter.Germ l R)

Equations

• Filter.Germ.commSemiring = CommSemiring.mk (_ : ∀ (a b : Filter.Germ l R), a * b = b * a)

instance Filter.Germ.nonUnitalCommRing {α : Type u_1} {l : Filter α} {R : Type u_5} [NonUnitalCommRing R] : NonUnitalCommRing (Filter.Germ l R)

Equations

• Filter.Germ.nonUnitalCommRing = let src := Filter.Germ.nonUnitalRing; let src_1 := Filter.Germ.commSemigroup; NonUnitalCommRing.mk (_ : ∀ (a b : Filter.Germ l R), a * b = b * a)

instance Filter.Germ.commRing {α : Type u_1} {l : Filter α} {R : Type u_5} [CommRing R] : CommRing (Filter.Germ l R)

Equations

• Filter.Germ.commRing = CommRing.mk (_ : ∀ (a b : Filter.Germ l R), a * b = b * a)

def Filter.Germ.coeRingHom {α : Type u_1} {R : Type u_5} [Semiring R] (l : Filter α) : (α → R) →+* Filter.Germ l R

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

Equations

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

@[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

instance Filter.Germ.instVAdd' {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [VAdd M β] : VAdd (Filter.Germ l M) (Filter.Germ l β)

Equations

• Filter.Germ.instVAdd' = { vadd := Filter.Germ.map₂ fun (x : M) (x_1 : β) => x +ᵥ x_1 }

instance Filter.Germ.instSMul' {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [SMul M β] : SMul (Filter.Germ l M) (Filter.Germ l β)

Equations

• Filter.Germ.instSMul' = { smul := Filter.Germ.map₂ fun (x : M) (x_1 : β) => x • x_1 }

@[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

@[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

instance Filter.Germ.addAction {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [AddMonoid M] [AddAction M β] : AddAction M (Filter.Germ l β)

Equations

• Filter.Germ.addAction = AddAction.mk (_ : ∀ (f : Filter.Germ l β), 0 +ᵥ f = f) (_ : ∀ (c₁ c₂ : M) (f : Filter.Germ l β), c₁ + c₂ +ᵥ f = c₁ +ᵥ (c₂ +ᵥ f))

theorem Filter.Germ.addAction.proof_1 {α : Type u_2} {β : Type u_1} {l : Filter α} {M : Type u_3} [AddMonoid M] [AddAction M β] (f : Filter.Germ l β) : 0 +ᵥ f = f

theorem Filter.Germ.addAction.proof_2 {α : Type u_2} {β : Type u_1} {l : Filter α} {M : Type u_3} [AddMonoid M] [AddAction M β] (c₁ : M) (c₂ : M) (f : Filter.Germ l β) : c₁ + c₂ +ᵥ f = c₁ +ᵥ (c₂ +ᵥ f)

instance Filter.Germ.mulAction {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [Monoid M] [MulAction M β] : MulAction M (Filter.Germ l β)

Equations

• Filter.Germ.mulAction = MulAction.mk (_ : ∀ (f : Filter.Germ l β), 1 • f = f) (_ : ∀ (c₁ c₂ : M) (f : Filter.Germ l β), (c₁ * c₂) • f = c₁ • c₂ • f)

theorem Filter.Germ.addAction'.proof_2 {α : Type u_2} {β : Type u_3} {l : Filter α} {M : Type u_1} [AddMonoid M] [AddAction M β] (c₁ : Filter.Germ l M) (c₂ : Filter.Germ l M) (f : Filter.Germ l β) : c₁ + c₂ +ᵥ f = c₁ +ᵥ (c₂ +ᵥ f)

instance Filter.Germ.addAction' {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [AddMonoid M] [AddAction M β] : AddAction (Filter.Germ l M) (Filter.Germ l β)

Equations

• Filter.Germ.addAction' = AddAction.mk (_ : ∀ (f : Filter.Germ l β), 0 +ᵥ f = f) (_ : ∀ (c₁ c₂ : Filter.Germ l M) (f : Filter.Germ l β), c₁ + c₂ +ᵥ f = c₁ +ᵥ (c₂ +ᵥ f))

theorem Filter.Germ.addAction'.proof_1 {α : Type u_2} {β : Type u_1} {l : Filter α} {M : Type u_3} [AddMonoid M] [AddAction M β] (f : Filter.Germ l β) : 0 +ᵥ f = f

instance Filter.Germ.mulAction' {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_5} [Monoid M] [MulAction M β] : MulAction (Filter.Germ l M) (Filter.Germ l β)

Equations

• Filter.Germ.mulAction' = MulAction.mk (_ : ∀ (f : Filter.Germ l β), 1 • f = f) (_ : ∀ (c₁ c₂ : Filter.Germ l M) (f : Filter.Germ l β), (c₁ * c₂) • f = c₁ • c₂ • f)

instance Filter.Germ.distribMulAction {α : Type u_1} {l : Filter α} {M : Type u_5} {N : Type u_6} [Monoid M] [AddMonoid N] [DistribMulAction M N] : DistribMulAction M (Filter.Germ l N)

Equations

• Filter.Germ.distribMulAction = DistribMulAction.mk (_ : ∀ (c : M), ↑(c • 0) = ↑0) (_ : ∀ (c : M) (f g : Filter.Germ l N), c • (f + g) = c • f + c • g)

instance Filter.Germ.distribMulAction' {α : Type u_1} {l : Filter α} {M : Type u_5} {N : Type u_6} [Monoid M] [AddMonoid N] [DistribMulAction M N] : DistribMulAction (Filter.Germ l M) (Filter.Germ l N)

Equations

• Filter.Germ.distribMulAction' = DistribMulAction.mk (_ : ∀ (c : Filter.Germ l M), c • 0 = 0) (_ : ∀ (c : Filter.Germ l M) (f g : Filter.Germ l N), c • (f + g) = c • f + c • g)

instance Filter.Germ.module {α : Type u_1} {l : Filter α} {M : Type u_5} {R : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] : Module R (Filter.Germ l M)

Equations

• Filter.Germ.module = Module.mk (_ : ∀ (c₁ c₂ : R) (f : Filter.Germ l M), (c₁ + c₂) • f = c₁ • f + c₂ • f) (_ : ∀ (f : Filter.Germ l M), 0 • f = 0)

instance Filter.Germ.module' {α : Type u_1} {l : Filter α} {M : Type u_5} {R : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] : Module (Filter.Germ l R) (Filter.Germ l M)

Equations

• Filter.Germ.module' = Module.mk (_ : ∀ (c₁ c₂ : Filter.Germ l R) (f : Filter.Germ l M), (c₁ + c₂) • f = c₁ • f + c₂ • f) (_ : ∀ (f : Filter.Germ l M), 0 • f = 0)

instance Filter.Germ.le {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] : LE (Filter.Germ l β)

Equations

• Filter.Germ.le = { le := Filter.Germ.LiftRel fun (x x_1 : β) => x ≤ x_1 }

theorem Filter.Germ.le_def {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] : (fun (x x_1 : Filter.Germ l β) => x ≤ x_1) = Filter.Germ.LiftRel fun (x x_1 : β) => x ≤ x_1

@[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

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

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

@[simp]

theorem Filter.Germ.const_le_iff {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] [Filter.NeBot l] {x : β} {y : β} : ↑x ≤ ↑y ↔ x ≤ y

instance Filter.Germ.preorder {α : Type u_1} {β : Type u_2} {l : Filter α} [Preorder β] : Preorder (Filter.Germ l β)

Equations

• Filter.Germ.preorder = Preorder.mk (_ : ∀ (f : Filter.Germ l β), f ≤ f) (_ : ∀ (f₁ f₂ f₃ : Filter.Germ l β), f₁ ≤ f₂ → f₂ ≤ f₃ → f₁ ≤ f₃)

instance Filter.Germ.partialOrder {α : Type u_1} {β : Type u_2} {l : Filter α} [PartialOrder β] : PartialOrder (Filter.Germ l β)

Equations

• Filter.Germ.partialOrder = let src := Filter.Germ.preorder; PartialOrder.mk (_ : ∀ (f g : Filter.Germ l β), f ≤ g → g ≤ f → f = g)

instance Filter.Germ.bot {α : Type u_1} {β : Type u_2} {l : Filter α} [Bot β] : Bot (Filter.Germ l β)

Equations

• Filter.Germ.bot = { bot := ↑⊥ }

instance Filter.Germ.top {α : Type u_1} {β : Type u_2} {l : Filter α} [Top β] : Top (Filter.Germ l β)

Equations

• Filter.Germ.top = { top := ↑⊤ }

@[simp]

theorem Filter.Germ.const_bot {α : Type u_1} {β : Type u_2} {l : Filter α} [Bot β] : ↑⊥ = ⊥

@[simp]

theorem Filter.Germ.const_top {α : Type u_1} {β : Type u_2} {l : Filter α} [Top β] : ↑⊤ = ⊤

instance Filter.Germ.orderBot {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] [OrderBot β] : OrderBot (Filter.Germ l β)

Equations

• Filter.Germ.orderBot = OrderBot.mk (_ : ∀ (f : Filter.Germ l β), ⊥ ≤ f)

instance Filter.Germ.orderTop {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] [OrderTop β] : OrderTop (Filter.Germ l β)

Equations

• Filter.Germ.orderTop = OrderTop.mk (_ : ∀ (f : Filter.Germ l β), f ≤ ⊤)

instance Filter.Germ.instBoundedOrderGermLe {α : Type u_1} {β : Type u_2} {l : Filter α} [LE β] [BoundedOrder β] : BoundedOrder (Filter.Germ l β)

Equations

• Filter.Germ.instBoundedOrderGermLe = let src := Filter.Germ.orderBot; let src_1 := Filter.Germ.orderTop; BoundedOrder.mk

instance Filter.Germ.sup {α : Type u_1} {β : Type u_2} {l : Filter α} [Sup β] : Sup (Filter.Germ l β)

Equations

• Filter.Germ.sup = { sup := Filter.Germ.map₂ fun (x x_1 : β) => x ⊔ x_1 }

instance Filter.Germ.inf {α : Type u_1} {β : Type u_2} {l : Filter α} [Inf β] : Inf (Filter.Germ l β)

Equations

• Filter.Germ.inf = { inf := Filter.Germ.map₂ fun (x x_1 : β) => x ⊓ x_1 }

@[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

@[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

instance Filter.Germ.semilatticeSup {α : Type u_1} {β : Type u_2} {l : Filter α} [SemilatticeSup β] : SemilatticeSup (Filter.Germ l β)

Equations

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

instance Filter.Germ.semilatticeInf {α : Type u_1} {β : Type u_2} {l : Filter α} [SemilatticeInf β] : SemilatticeInf (Filter.Germ l β)

Equations

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

instance Filter.Germ.lattice {α : Type u_1} {β : Type u_2} {l : Filter α} [Lattice β] : Lattice (Filter.Germ l β)

Equations

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

instance Filter.Germ.distribLattice {α : Type u_1} {β : Type u_2} {l : Filter α} [DistribLattice β] : DistribLattice (Filter.Germ l β)

Equations

• Filter.Germ.distribLattice = let src := Filter.Germ.semilatticeSup; let src_1 := Filter.Germ.semilatticeInf; DistribLattice.mk (_ : ∀ (f g h : Filter.Germ l β), (f ⊔ g) ⊓ (f ⊔ h) ≤ f ⊔ g ⊓ h)

theorem Filter.Germ.orderedAddCommMonoid.proof_1 {α : Type u_2} {β : Type u_1} {l : Filter α} [OrderedAddCommMonoid β] (f : Filter.Germ l β) (g : Filter.Germ l β) : f ≤ g → ∀ (c : Filter.Germ l β), c + f ≤ c + g

instance Filter.Germ.orderedAddCommMonoid {α : Type u_1} {β : Type u_2} {l : Filter α} [OrderedAddCommMonoid β] : OrderedAddCommMonoid (Filter.Germ l β)

Equations

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

instance Filter.Germ.orderedCommMonoid {α : Type u_1} {β : Type u_2} {l : Filter α} [OrderedCommMonoid β] : OrderedCommMonoid (Filter.Germ l β)

Equations

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

instance Filter.Germ.orderedAddCancelCommMonoid {α : Type u_1} {β : Type u_2} {l : Filter α} [OrderedCancelAddCommMonoid β] : OrderedCancelAddCommMonoid (Filter.Germ l β)

Equations

• Filter.Germ.orderedAddCancelCommMonoid = let src := Filter.Germ.orderedAddCommMonoid; OrderedCancelAddCommMonoid.mk (_ : ∀ (f g h : Filter.Germ l β), f + g ≤ f + h → g ≤ h)

theorem Filter.Germ.orderedAddCancelCommMonoid.proof_1 {α : Type u_2} {β : Type u_1} {l : Filter α} [OrderedCancelAddCommMonoid β] (f : Filter.Germ l β) (g : Filter.Germ l β) (h : Filter.Germ l β) : f + g ≤ f + h → g ≤ h

instance Filter.Germ.orderedCancelCommMonoid {α : Type u_1} {β : Type u_2} {l : Filter α} [OrderedCancelCommMonoid β] : OrderedCancelCommMonoid (Filter.Germ l β)

Equations

• Filter.Germ.orderedCancelCommMonoid = let src := Filter.Germ.orderedCommMonoid; OrderedCancelCommMonoid.mk (_ : ∀ (f g h : Filter.Germ l β), f * g ≤ f * h → g ≤ h)

theorem Filter.Germ.orderedAddCommGroup.proof_4 {α : Type u_2} {β : Type u_1} {l : Filter α} [OrderedAddCommGroup β] (n : ℕ) (a : Filter.Germ l β) : SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑(Nat.succ n)) a

theorem Filter.Germ.orderedAddCommGroup.proof_1 {α : Type u_2} {β : Type u_1} {l : Filter α} [OrderedAddCommGroup β] (a : Filter.Germ l β) (b : Filter.Germ l β) : a - b = a + -b

theorem Filter.Germ.orderedAddCommGroup.proof_5 {α : Type u_2} {β : Type u_1} {l : Filter α} [OrderedAddCommGroup β] (a : Filter.Germ l β) : -a + a = 0

theorem Filter.Germ.orderedAddCommGroup.proof_2 {α : Type u_2} {β : Type u_1} {l : Filter α} [OrderedAddCommGroup β] (a : Filter.Germ l β) : SubNegMonoid.zsmul 0 a = 0

theorem Filter.Germ.orderedAddCommGroup.proof_7 {α : Type u_2} {β : Type u_1} {l : Filter α} [OrderedAddCommGroup β] (a : Filter.Germ l β) (b : Filter.Germ l β) : a ≤ b → ∀ (c : Filter.Germ l β), c + a ≤ c + b

instance Filter.Germ.orderedAddCommGroup {α : Type u_1} {β : Type u_2} {l : Filter α} [OrderedAddCommGroup β] : OrderedAddCommGroup (Filter.Germ l β)

Equations

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

theorem Filter.Germ.orderedAddCommGroup.proof_6 {α : Type u_2} {β : Type u_1} {l : Filter α} [OrderedAddCommGroup β] (a : Filter.Germ l β) (b : Filter.Germ l β) : a + b = b + a

theorem Filter.Germ.orderedAddCommGroup.proof_3 {α : Type u_2} {β : Type u_1} {l : Filter α} [OrderedAddCommGroup β] (n : ℕ) (a : Filter.Germ l β) : SubNegMonoid.zsmul (Int.ofNat (Nat.succ n)) a = a + SubNegMonoid.zsmul (Int.ofNat n) a

instance Filter.Germ.orderedCommGroup {α : Type u_1} {β : Type u_2} {l : Filter α} [OrderedCommGroup β] : OrderedCommGroup (Filter.Germ l β)

Equations

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

instance Filter.Germ.existsAddOfLE {α : Type u_1} {β : Type u_2} {l : Filter α} [Add β] [LE β] [ExistsAddOfLE β] : ExistsAddOfLE (Filter.Germ l β)

Equations

• (_ : ExistsAddOfLE (Filter.Germ l β)) = (_ : ExistsAddOfLE (Filter.Germ l β))

instance Filter.Germ.existsMulOfLE {α : Type u_1} {β : Type u_2} {l : Filter α} [Mul β] [LE β] [ExistsMulOfLE β] : ExistsMulOfLE (Filter.Germ l β)

Equations

• (_ : ExistsMulOfLE (Filter.Germ l β)) = (_ : ExistsMulOfLE (Filter.Germ l β))

instance Filter.Germ.CanonicallyOrderedAddCommMonoid {α : Type u_1} {β : Type u_2} {l : Filter α} [CanonicallyOrderedAddCommMonoid β] : CanonicallyOrderedAddCommMonoid (Filter.Germ l β)

Equations

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

theorem Filter.Germ.CanonicallyOrderedAddCommMonoid.proof_1 {α : Type u_1} {β : Type u_2} {l : Filter α} [CanonicallyOrderedAddCommMonoid β] : ExistsAddOfLE (Filter.Germ l β)

theorem Filter.Germ.CanonicallyOrderedAddCommMonoid.proof_2 {α : Type u_2} {β : Type u_1} {l : Filter α} [CanonicallyOrderedAddCommMonoid β] : ∀ {a b : Filter.Germ l β}, a ≤ b → ∃ (c : Filter.Germ l β), b = a + c

theorem Filter.Germ.CanonicallyOrderedAddCommMonoid.proof_3 {α : Type u_2} {β : Type u_1} {l : Filter α} [CanonicallyOrderedAddCommMonoid β] (x : Filter.Germ l β) (y : Filter.Germ l β) : x ≤ x + y

instance Filter.Germ.CanonicallyOrderedCommMonoid {α : Type u_1} {β : Type u_2} {l : Filter α} [CanonicallyOrderedCommMonoid β] : CanonicallyOrderedCommMonoid (Filter.Germ l β)

Equations

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

instance Filter.Germ.orderedSemiring {α : Type u_1} {β : Type u_2} {l : Filter α} [OrderedSemiring β] : OrderedSemiring (Filter.Germ l β)

Equations

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

instance Filter.Germ.orderedCommSemiring {α : Type u_1} {β : Type u_2} {l : Filter α} [OrderedCommSemiring β] : OrderedCommSemiring (Filter.Germ l β)

Equations

• Filter.Germ.orderedCommSemiring = let src := Filter.Germ.orderedSemiring; let src_1 := Filter.Germ.commSemiring; OrderedCommSemiring.mk (_ : ∀ (a b : Filter.Germ l β), a * b = b * a)

instance Filter.Germ.orderedRing {α : Type u_1} {β : Type u_2} {l : Filter α} [OrderedRing β] : OrderedRing (Filter.Germ l β)

Equations

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

instance Filter.Germ.orderedCommRing {α : Type u_1} {β : Type u_2} {l : Filter α} [OrderedCommRing β] : OrderedCommRing (Filter.Germ l β)

Equations

• Filter.Germ.orderedCommRing = let src := Filter.Germ.orderedRing; let src_1 := Filter.Germ.orderedCommSemiring; OrderedCommRing.mk (_ : ∀ (a b : Filter.Germ l β), a * b = b * a)