File: Mathlib/Analysis/Asymptotics/Theta.lean

full_name: Asymptotics.IsTheta
code: /-- We say that `f` is `Θ(g)` along a filter `l` (notation: `f =Θ[l] g`) if `f =O[l] g` and
`g =O[l] f`. -/
def IsTheta (l : Filter α) (f : α → E) (g : α → F) : Prop 
goal: Prop

full_name: Asymptotics.«_aux_Mathlib_Analysis_Asymptotics_Theta___macroRules_Asymptotics_term_=Θ[_]__1»
code: @[inherit_doc]
notation:100 f " =Θ[" l "] " g:100 => IsTheta l f g
goal: Lean.Macro

full_name: Asymptotics._aux_Mathlib_Analysis_Asymptotics_Theta___unexpand_Asymptotics_IsTheta_1
code: @[inherit_doc]
notation:100 f " =Θ[" l "] " g:100 => IsTheta l f g
goal: Lean.PrettyPrinter.Unexpander

full_name: Asymptotics.IsBigO.antisymm
code: theorem IsBigO.antisymm (h₁ : f =O[l] g) (h₂ : g =O[l] f) : f =Θ[l] g 
goal: f =Θ[l] g

full_name: Asymptotics.IsTheta.isBigO
code: lemma IsTheta.isBigO (h : f =Θ[l] g) : f =O[l] g 
goal: f =O[l] g

full_name: Asymptotics.IsTheta.isBigO_symm
code: lemma IsTheta.isBigO_symm (h : f =Θ[l] g) : g =O[l] f 
goal: g =O[l] f

full_name: Asymptotics.isTheta_refl
code: @[refl]
theorem isTheta_refl (f : α → E) (l : Filter α) : f =Θ[l] f 
goal: f =Θ[l] f

full_name: Asymptotics.isTheta_rfl
code: theorem isTheta_rfl : f =Θ[l] f 
goal: f =Θ[l] f

full_name: Asymptotics.IsTheta.symm
code: @[symm]
nonrec theorem IsTheta.symm (h : f =Θ[l] g) : g =Θ[l] f 
goal: g =Θ[l] f

full_name: Asymptotics.isTheta_comm
code: theorem isTheta_comm : f =Θ[l] g ↔ g =Θ[l] f 
goal: f =Θ[l] g ↔ g =Θ[l] f

full_name: Asymptotics.IsTheta.trans
code: @[trans]
theorem IsTheta.trans {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =Θ[l] g) (h₂ : g =Θ[l] k) :
    f =Θ[l] k 
goal: f =Θ[l] k

full_name: Asymptotics.instTransForallIsTheta
code: instance : Trans (α := α → E) (β := α → F') (γ := α → G) (IsTheta l) (IsTheta l) (IsTheta l) 
goal: Trans (Asymptotics.IsTheta l) (Asymptotics.IsTheta l) (Asymptotics.IsTheta l)

full_name: Asymptotics.IsBigO.trans_isTheta
code: @[trans]
theorem IsBigO.trans_isTheta {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =O[l] g)
    (h₂ : g =Θ[l] k) : f =O[l] k 
goal: f =O[l] k

full_name: Asymptotics.instTransForallIsBigOIsTheta
code: instance : Trans (α := α → E) (β := α → F') (γ := α → G) (IsBigO l) (IsTheta l) (IsBigO l) 
goal: Trans (Asymptotics.IsBigO l) (Asymptotics.IsTheta l) (Asymptotics.IsBigO l)

full_name: Asymptotics.IsTheta.trans_isBigO
code: @[trans]
theorem IsTheta.trans_isBigO {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =Θ[l] g)
    (h₂ : g =O[l] k) : f =O[l] k 
goal: f =O[l] k

full_name: Asymptotics.instTransForallIsThetaIsBigO
code: instance : Trans (α := α → E) (β := α → F') (γ := α → G) (IsTheta l) (IsBigO l) (IsBigO l) 
goal: Trans (Asymptotics.IsTheta l) (Asymptotics.IsBigO l) (Asymptotics.IsBigO l)

full_name: Asymptotics.IsLittleO.trans_isTheta
code: @[trans]
theorem IsLittleO.trans_isTheta {f : α → E} {g : α → F} {k : α → G'} (h₁ : f =o[l] g)
    (h₂ : g =Θ[l] k) : f =o[l] k 
goal: f =o[l] k

full_name: Asymptotics.instTransForallIsLittleOIsTheta
code: instance : Trans (α := α → E) (β := α → F') (γ := α → G') (IsLittleO l) (IsTheta l) (IsLittleO l) 
goal: Trans (Asymptotics.IsLittleO l) (Asymptotics.IsTheta l) (Asymptotics.IsLittleO l)

full_name: Asymptotics.IsTheta.trans_isLittleO
code: @[trans]
theorem IsTheta.trans_isLittleO {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =Θ[l] g)
    (h₂ : g =o[l] k) : f =o[l] k 
goal: f =o[l] k

full_name: Asymptotics.instTransForallIsThetaIsLittleO
code: instance : Trans (α := α → E) (β := α → F') (γ := α → G) (IsTheta l) (IsLittleO l) (IsLittleO l) 
goal: Trans (Asymptotics.IsTheta l) (Asymptotics.IsLittleO l) (Asymptotics.IsLittleO l)

full_name: Asymptotics.IsTheta.trans_eventuallyEq
code: @[trans]
theorem IsTheta.trans_eventuallyEq {f : α → E} {g₁ g₂ : α → F} (h : f =Θ[l] g₁) (hg : g₁ =ᶠ[l] g₂) :
    f =Θ[l] g₂ 
goal: f =Θ[l] g₂

full_name: Asymptotics.instTransForallIsThetaEventuallyEq
code: instance : Trans (α := α → E) (β := α → F) (γ := α → F) (IsTheta l) (EventuallyEq l) (IsTheta l) 
goal: Trans (Asymptotics.IsTheta l) l.EventuallyEq (Asymptotics.IsTheta l)

full_name: Filter.EventuallyEq.trans_isTheta
code: @[trans]
theorem _root_.Filter.EventuallyEq.trans_isTheta {f₁ f₂ : α → E} {g : α → F} (hf : f₁ =ᶠ[l] f₂)
    (h : f₂ =Θ[l] g) : f₁ =Θ[l] g 
goal: f₁ =Θ[l] g

full_name: Asymptotics.instTransForallEventuallyEqIsTheta
code: instance : Trans (α := α → E) (β := α → E) (γ := α → F) (EventuallyEq l) (IsTheta l) (IsTheta l) 
goal: Trans l.EventuallyEq (Asymptotics.IsTheta l) (Asymptotics.IsTheta l)

full_name: Filter.EventuallyEq.isTheta
code: lemma _root_.Filter.EventuallyEq.isTheta {f g : α → E} (h : f =ᶠ[l] g) : f =Θ[l] g 
goal: f =Θ[l] g

full_name: Asymptotics.isTheta_norm_left
code: @[simp]
theorem isTheta_norm_left : (fun x ↦ ‖f' x‖) =Θ[l] g ↔ f' =Θ[l] g 
goal: (fun x => ‖f' x‖) =Θ[l] g ↔ f' =Θ[l] g

full_name: Asymptotics.isTheta_norm_right
code: @[simp]
theorem isTheta_norm_right : (f =Θ[l] fun x ↦ ‖g' x‖) ↔ f =Θ[l] g' 
goal: (f =Θ[l] fun x => ‖g' x‖) ↔ f =Θ[l] g'

full_name: Asymptotics.isTheta_of_norm_eventuallyEq
code: theorem isTheta_of_norm_eventuallyEq (h : (fun x ↦ ‖f x‖) =ᶠ[l] fun x ↦ ‖g x‖) : f =Θ[l] g 
goal: f =Θ[l] g

full_name: Asymptotics.isTheta_of_norm_eventuallyEq'
code: theorem isTheta_of_norm_eventuallyEq' {g : α → ℝ} (h : (fun x ↦ ‖f' x‖) =ᶠ[l] g) : f' =Θ[l] g 
goal: f' =Θ[l] g

full_name: Asymptotics.IsTheta.isLittleO_congr_left
code: theorem IsTheta.isLittleO_congr_left (h : f' =Θ[l] g') : f' =o[l] k ↔ g' =o[l] k 
goal: f' =o[l] k ↔ g' =o[l] k

full_name: Asymptotics.IsTheta.isLittleO_congr_right
code: theorem IsTheta.isLittleO_congr_right (h : g' =Θ[l] k') : f =o[l] g' ↔ f =o[l] k' 
goal: f =o[l] g' ↔ f =o[l] k'

full_name: Asymptotics.IsTheta.isBigO_congr_left
code: theorem IsTheta.isBigO_congr_left (h : f' =Θ[l] g') : f' =O[l] k ↔ g' =O[l] k 
goal: f' =O[l] k ↔ g' =O[l] k

full_name: Asymptotics.IsTheta.isBigO_congr_right
code: theorem IsTheta.isBigO_congr_right (h : g' =Θ[l] k') : f =O[l] g' ↔ f =O[l] k' 
goal: f =O[l] g' ↔ f =O[l] k'

full_name: Asymptotics.IsTheta.isTheta_congr_left
code: lemma IsTheta.isTheta_congr_left (h : f' =Θ[l] g') : f' =Θ[l] k ↔ g' =Θ[l] k 
goal: f' =Θ[l] k ↔ g' =Θ[l] k

full_name: Asymptotics.IsTheta.isTheta_congr_right
code: lemma IsTheta.isTheta_congr_right (h : f' =Θ[l] g') : k =Θ[l] f' ↔ k =Θ[l] g' 
goal: k =Θ[l] f' ↔ k =Θ[l] g'

full_name: Asymptotics.IsTheta.mono
code: theorem IsTheta.mono (h : f =Θ[l] g) (hl : l' ≤ l) : f =Θ[l'] g 
goal: f =Θ[l'] g

full_name: Asymptotics.IsTheta.sup
code: theorem IsTheta.sup (h : f' =Θ[l] g') (h' : f' =Θ[l'] g') : f' =Θ[l ⊔ l'] g' 
goal: f' =Θ[l ⊔ l'] g'

full_name: Asymptotics.isTheta_sup
code: @[simp]
theorem isTheta_sup : f' =Θ[l ⊔ l'] g' ↔ f' =Θ[l] g' ∧ f' =Θ[l'] g' 
goal: f' =Θ[l ⊔ l'] g' ↔ f' =Θ[l] g' ∧ f' =Θ[l'] g'

full_name: Asymptotics.IsTheta.eq_zero_iff
code: theorem IsTheta.eq_zero_iff (h : f'' =Θ[l] g'') : ∀ᶠ x in l, f'' x = 0 ↔ g'' x = 0 
goal: ∀ᶠ (x : α) in l, f'' x = 0 ↔ g'' x = 0

full_name: Asymptotics.IsTheta.tendsto_zero_iff
code: theorem IsTheta.tendsto_zero_iff (h : f'' =Θ[l] g'') :
    Tendsto f'' l (𝓝 0) ↔ Tendsto g'' l (𝓝 0) 
goal: Filter.Tendsto f'' l (𝓝 0) ↔ Filter.Tendsto g'' l (𝓝 0)

full_name: Asymptotics.IsTheta.tendsto_norm_atTop_iff
code: theorem IsTheta.tendsto_norm_atTop_iff (h : f' =Θ[l] g') :
    Tendsto (norm ∘ f') l atTop ↔ Tendsto (norm ∘ g') l atTop 
goal: Filter.Tendsto (norm ∘ f') l Filter.atTop ↔ Filter.Tendsto (norm ∘ g') l Filter.atTop

full_name: Asymptotics.IsTheta.isBoundedUnder_le_iff
code: theorem IsTheta.isBoundedUnder_le_iff (h : f' =Θ[l] g') :
    IsBoundedUnder (· ≤ ·) l (norm ∘ f') ↔ IsBoundedUnder (· ≤ ·) l (norm ∘ g') 
goal: Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f') ↔ Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ g')

full_name: Asymptotics.IsTheta.smul
code: theorem IsTheta.smul [NormedSpace 𝕜 E'] [NormedSpace 𝕜' F'] {f₁ : α → 𝕜} {f₂ : α → 𝕜'} {g₁ : α → E'}
    {g₂ : α → F'} (hf : f₁ =Θ[l] f₂) (hg : g₁ =Θ[l] g₂) :
    (fun x ↦ f₁ x • g₁ x) =Θ[l] fun x ↦ f₂ x • g₂ x 
goal: (fun x => f₁ x • g₁ x) =Θ[l] fun x => f₂ x • g₂ x

full_name: Asymptotics.IsTheta.mul
code: theorem IsTheta.mul {f₁ f₂ : α → 𝕜} {g₁ g₂ : α → 𝕜'} (h₁ : f₁ =Θ[l] g₁) (h₂ : f₂ =Θ[l] g₂) :
    (fun x ↦ f₁ x * f₂ x) =Θ[l] fun x ↦ g₁ x * g₂ x 
goal: (fun x => f₁ x * f₂ x) =Θ[l] fun x => g₁ x * g₂ x

full_name: Asymptotics.IsTheta.inv
code: theorem IsTheta.inv {f : α → 𝕜} {g : α → 𝕜'} (h : f =Θ[l] g) :
    (fun x ↦ (f x)⁻¹) =Θ[l] fun x ↦ (g x)⁻¹ 
goal: (fun x => (f x)⁻¹) =Θ[l] fun x => (g x)⁻¹

full_name: Asymptotics.isTheta_inv
code: @[simp]
theorem isTheta_inv {f : α → 𝕜} {g : α → 𝕜'} :
    ((fun x ↦ (f x)⁻¹) =Θ[l] fun x ↦ (g x)⁻¹) ↔ f =Θ[l] g 
goal: ((fun x => (f x)⁻¹) =Θ[l] fun x => (g x)⁻¹) ↔ f =Θ[l] g

full_name: Asymptotics.IsTheta.div
code: theorem IsTheta.div {f₁ f₂ : α → 𝕜} {g₁ g₂ : α → 𝕜'} (h₁ : f₁ =Θ[l] g₁) (h₂ : f₂ =Θ[l] g₂) :
    (fun x ↦ f₁ x / f₂ x) =Θ[l] fun x ↦ g₁ x / g₂ x 
goal: (fun x => f₁ x / f₂ x) =Θ[l] fun x => g₁ x / g₂ x

full_name: Asymptotics.IsTheta.pow
code: theorem IsTheta.pow {f : α → 𝕜} {g : α → 𝕜'} (h : f =Θ[l] g) (n : ℕ) :
    (fun x ↦ f x ^ n) =Θ[l] fun x ↦ g x ^ n 
goal: (fun x => f x ^ n) =Θ[l] fun x => g x ^ n

full_name: Asymptotics.IsTheta.zpow
code: theorem IsTheta.zpow {f : α → 𝕜} {g : α → 𝕜'} (h : f =Θ[l] g) (n : ℤ) :
    (fun x ↦ f x ^ n) =Θ[l] fun x ↦ g x ^ n 
goal: (fun x => f x ^ n) =Θ[l] fun x => g x ^ n

full_name: Asymptotics.isTheta_const_const
code: theorem isTheta_const_const {c₁ : E''} {c₂ : F''} (h₁ : c₁ ≠ 0) (h₂ : c₂ ≠ 0) :
    (fun _ : α ↦ c₁) =Θ[l] fun _ ↦ c₂ 
goal: (fun x => c₁) =Θ[l] fun x => c₂

full_name: Asymptotics.isTheta_const_const_iff
code: @[simp]
theorem isTheta_const_const_iff [NeBot l] {c₁ : E''} {c₂ : F''} :
    ((fun _ : α ↦ c₁) =Θ[l] fun _ ↦ c₂) ↔ (c₁ = 0 ↔ c₂ = 0) 
goal: ((fun x => c₁) =Θ[l] fun x => c₂) ↔ (c₁ = 0 ↔ c₂ = 0)

full_name: Asymptotics.isTheta_zero_left
code: @[simp]
theorem isTheta_zero_left : (fun _ ↦ (0 : E')) =Θ[l] g'' ↔ g'' =ᶠ[l] 0 
goal: (fun x => 0) =Θ[l] g'' ↔ g'' =ᶠ[l] 0

full_name: Asymptotics.isTheta_zero_right
code: @[simp]
theorem isTheta_zero_right : (f'' =Θ[l] fun _ ↦ (0 : F')) ↔ f'' =ᶠ[l] 0 
goal: (f'' =Θ[l] fun x => 0) ↔ f'' =ᶠ[l] 0

full_name: Asymptotics.isTheta_const_smul_left
code: theorem isTheta_const_smul_left [NormedSpace 𝕜 E'] {c : 𝕜} (hc : c ≠ 0) :
    (fun x ↦ c • f' x) =Θ[l] g ↔ f' =Θ[l] g 
goal: (fun x => c • f' x) =Θ[l] g ↔ f' =Θ[l] g

full_name: Asymptotics.isTheta_const_smul_right
code: theorem isTheta_const_smul_right [NormedSpace 𝕜 F'] {c : 𝕜} (hc : c ≠ 0) :
    (f =Θ[l] fun x ↦ c • g' x) ↔ f =Θ[l] g' 
goal: (f =Θ[l] fun x => c • g' x) ↔ f =Θ[l] g'

full_name: Asymptotics.isTheta_const_mul_left
code: theorem isTheta_const_mul_left {c : 𝕜} {f : α → 𝕜} (hc : c ≠ 0) :
    (fun x ↦ c * f x) =Θ[l] g ↔ f =Θ[l] g 
goal: (fun x => c * f x) =Θ[l] g ↔ f =Θ[l] g

full_name: Asymptotics.isTheta_const_mul_right
code: theorem isTheta_const_mul_right {c : 𝕜} {g : α → 𝕜} (hc : c ≠ 0) :
    (f =Θ[l] fun x ↦ c * g x) ↔ f =Θ[l] g 
goal: (f =Θ[l] fun x => c * g x) ↔ f =Θ[l] g

full_name: Asymptotics.IsLittleO.right_isTheta_add
code: theorem IsLittleO.right_isTheta_add {f₁ f₂ : α → E'} (h : f₁ =o[l] f₂) :
    f₂ =Θ[l] (f₁ + f₂) 
goal: f₂ =Θ[l] (f₁ + f₂)

full_name: Asymptotics.IsLittleO.right_isTheta_add'
code: theorem IsLittleO.right_isTheta_add' {f₁ f₂ : α → E'} (h : f₁ =o[l] f₂) :
    f₂ =Θ[l] (f₂ + f₁) 
goal: f₂ =Θ[l] (f₂ + f₁)

full_name: Asymptotics.IsTheta.add_isLittleO
code: lemma IsTheta.add_isLittleO {f₁ f₂ : α → E'} {g : α → F}
    (hΘ : f₁ =Θ[l] g) (ho : f₂ =o[l] g) : (f₁ + f₂) =Θ[l] g 
goal: (f₁ + f₂) =Θ[l] g

full_name: Asymptotics.IsLittleO.add_isTheta
code: lemma IsLittleO.add_isTheta {f₁ f₂ : α → E'} {g : α → F}
    (ho : f₁ =o[l] g) (hΘ : f₂ =Θ[l] g) : (f₁ + f₂) =Θ[l] g 
goal: (f₁ + f₂) =Θ[l] g