File: Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean

full_name: Asymptotics.SuperpolynomialDecay
code: /-- `f` has superpolynomial decay in parameter `k` along filter `l` if
  `k ^ n * f` tends to zero at `l` for all naturals `n` -/
def SuperpolynomialDecay {α β : Type*} [TopologicalSpace β] [CommSemiring β] (l : Filter α)
    (k : α → β) (f : α → β) 
goal: Prop

full_name: Asymptotics.SuperpolynomialDecay.congr'
code: theorem SuperpolynomialDecay.congr' (hf : SuperpolynomialDecay l k f) (hfg : f =ᶠ[l] g) :
    SuperpolynomialDecay l k g 
goal: Asymptotics.SuperpolynomialDecay l k g

full_name: Asymptotics.SuperpolynomialDecay.congr
code: theorem SuperpolynomialDecay.congr (hf : SuperpolynomialDecay l k f) (hfg : ∀ x, f x = g x) :
    SuperpolynomialDecay l k g 
goal: Asymptotics.SuperpolynomialDecay l k g

full_name: Asymptotics.superpolynomialDecay_zero
code: @[simp]
theorem superpolynomialDecay_zero (l : Filter α) (k : α → β) : SuperpolynomialDecay l k 0 
goal: Asymptotics.SuperpolynomialDecay l k 0

full_name: Asymptotics.SuperpolynomialDecay.add
code: theorem SuperpolynomialDecay.add [ContinuousAdd β] (hf : SuperpolynomialDecay l k f)
    (hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f + g) 
goal: Asymptotics.SuperpolynomialDecay l k (f + g)

full_name: Asymptotics.SuperpolynomialDecay.mul
code: theorem SuperpolynomialDecay.mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f)
    (hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f * g) 
goal: Asymptotics.SuperpolynomialDecay l k (f * g)

full_name: Asymptotics.SuperpolynomialDecay.mul_const
code: theorem SuperpolynomialDecay.mul_const [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :
    SuperpolynomialDecay l k fun n => f n * c 
goal: Asymptotics.SuperpolynomialDecay l k fun n => f n * c

full_name: Asymptotics.SuperpolynomialDecay.const_mul
code: theorem SuperpolynomialDecay.const_mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :
    SuperpolynomialDecay l k fun n => c * f n 
goal: Asymptotics.SuperpolynomialDecay l k fun n => c * f n

full_name: Asymptotics.SuperpolynomialDecay.param_mul
code: theorem SuperpolynomialDecay.param_mul (hf : SuperpolynomialDecay l k f) :
    SuperpolynomialDecay l k (k * f) 
goal: Asymptotics.SuperpolynomialDecay l k (k * f)

full_name: Asymptotics.SuperpolynomialDecay.mul_param
code: theorem SuperpolynomialDecay.mul_param (hf : SuperpolynomialDecay l k f) :
    SuperpolynomialDecay l k (f * k) 
goal: Asymptotics.SuperpolynomialDecay l k (f * k)

full_name: Asymptotics.SuperpolynomialDecay.param_pow_mul
code: theorem SuperpolynomialDecay.param_pow_mul (hf : SuperpolynomialDecay l k f) (n : ℕ) :
    SuperpolynomialDecay l k (k ^ n * f) 
goal: Asymptotics.SuperpolynomialDecay l k (k ^ n * f)

full_name: Asymptotics.SuperpolynomialDecay.mul_param_pow
code: theorem SuperpolynomialDecay.mul_param_pow (hf : SuperpolynomialDecay l k f) (n : ℕ) :
    SuperpolynomialDecay l k (f * k ^ n) 
goal: Asymptotics.SuperpolynomialDecay l k (f * k ^ n)

full_name: Asymptotics.SuperpolynomialDecay.polynomial_mul
code: theorem SuperpolynomialDecay.polynomial_mul [ContinuousAdd β] [ContinuousMul β]
    (hf : SuperpolynomialDecay l k f) (p : β[X]) :
    SuperpolynomialDecay l k fun x => (p.eval <| k x) * f x 
goal: Asymptotics.SuperpolynomialDecay l k fun x => Polynomial.eval (k x) p * f x

full_name: Asymptotics.SuperpolynomialDecay.mul_polynomial
code: theorem SuperpolynomialDecay.mul_polynomial [ContinuousAdd β] [ContinuousMul β]
    (hf : SuperpolynomialDecay l k f) (p : β[X]) :
    SuperpolynomialDecay l k fun x => f x * (p.eval <| k x) 
goal: Asymptotics.SuperpolynomialDecay l k fun x => f x * Polynomial.eval (k x) p

full_name: Asymptotics.SuperpolynomialDecay.trans_eventuallyLE
code: theorem SuperpolynomialDecay.trans_eventuallyLE (hk : 0 ≤ᶠ[l] k) (hg : SuperpolynomialDecay l k g)
    (hg' : SuperpolynomialDecay l k g') (hfg : g ≤ᶠ[l] f) (hfg' : f ≤ᶠ[l] g') :
    SuperpolynomialDecay l k f 
goal: Asymptotics.SuperpolynomialDecay l k f

full_name: Asymptotics.superpolynomialDecay_iff_abs_tendsto_zero
code: theorem superpolynomialDecay_iff_abs_tendsto_zero :
    SuperpolynomialDecay l k f ↔ ∀ n : ℕ, Tendsto (fun a : α => |k a ^ n * f a|) l (𝓝 0) 
goal: Asymptotics.SuperpolynomialDecay l k f ↔ ∀ (n : ℕ), Filter.Tendsto (fun a => |k a ^ n * f a|) l (nhds 0)

full_name: Asymptotics.superpolynomialDecay_iff_superpolynomialDecay_abs
code: theorem superpolynomialDecay_iff_superpolynomialDecay_abs :
    SuperpolynomialDecay l k f ↔ SuperpolynomialDecay l (fun a => |k a|) fun a => |f a| 
goal: Asymptotics.SuperpolynomialDecay l k f ↔ Asymptotics.SuperpolynomialDecay l (fun a => |k a|) fun a => |f a|

full_name: Asymptotics.SuperpolynomialDecay.trans_eventually_abs_le
code: theorem SuperpolynomialDecay.trans_eventually_abs_le (hf : SuperpolynomialDecay l k f)
    (hfg : abs ∘ g ≤ᶠ[l] abs ∘ f) : SuperpolynomialDecay l k g 
goal: Asymptotics.SuperpolynomialDecay l k g

full_name: Asymptotics.SuperpolynomialDecay.trans_abs_le
code: theorem SuperpolynomialDecay.trans_abs_le (hf : SuperpolynomialDecay l k f)
    (hfg : ∀ x, |g x| ≤ |f x|) : SuperpolynomialDecay l k g 
goal: Asymptotics.SuperpolynomialDecay l k g

full_name: Asymptotics.superpolynomialDecay_mul_const_iff
code: theorem superpolynomialDecay_mul_const_iff [ContinuousMul β] {c : β} (hc0 : c ≠ 0) :
    (SuperpolynomialDecay l k fun n => f n * c) ↔ SuperpolynomialDecay l k f 
goal: (Asymptotics.SuperpolynomialDecay l k fun n => f n * c) ↔ Asymptotics.SuperpolynomialDecay l k f

full_name: Asymptotics.superpolynomialDecay_const_mul_iff
code: theorem superpolynomialDecay_const_mul_iff [ContinuousMul β] {c : β} (hc0 : c ≠ 0) :
    (SuperpolynomialDecay l k fun n => c * f n) ↔ SuperpolynomialDecay l k f 
goal: (Asymptotics.SuperpolynomialDecay l k fun n => c * f n) ↔ Asymptotics.SuperpolynomialDecay l k f

full_name: Asymptotics.superpolynomialDecay_iff_abs_isBoundedUnder
code: theorem superpolynomialDecay_iff_abs_isBoundedUnder (hk : Tendsto k l atTop) :
    SuperpolynomialDecay l k f ↔
    ∀ z : ℕ, IsBoundedUnder (· ≤ ·) l fun a : α => |k a ^ z * f a| 
goal: Asymptotics.SuperpolynomialDecay l k f ↔
  ∀ (z : ℕ), Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => |k a ^ z * f a|

full_name: Asymptotics.superpolynomialDecay_iff_zpow_tendsto_zero
code: theorem superpolynomialDecay_iff_zpow_tendsto_zero (hk : Tendsto k l atTop) :
    SuperpolynomialDecay l k f ↔ ∀ z : ℤ, Tendsto (fun a : α => k a ^ z * f a) l (𝓝 0) 
goal: Asymptotics.SuperpolynomialDecay l k f ↔ ∀ (z : ℤ), Filter.Tendsto (fun a => k a ^ z * f a) l (nhds 0)

full_name: Asymptotics.SuperpolynomialDecay.param_zpow_mul
code: theorem SuperpolynomialDecay.param_zpow_mul (hk : Tendsto k l atTop)
    (hf : SuperpolynomialDecay l k f) (z : ℤ) :
    SuperpolynomialDecay l k fun a => k a ^ z * f a 
goal: Asymptotics.SuperpolynomialDecay l k fun a => k a ^ z * f a

full_name: Asymptotics.SuperpolynomialDecay.mul_param_zpow
code: theorem SuperpolynomialDecay.mul_param_zpow (hk : Tendsto k l atTop)
    (hf : SuperpolynomialDecay l k f) (z : ℤ) : SuperpolynomialDecay l k fun a => f a * k a ^ z 
goal: Asymptotics.SuperpolynomialDecay l k fun a => f a * k a ^ z

full_name: Asymptotics.SuperpolynomialDecay.inv_param_mul
code: theorem SuperpolynomialDecay.inv_param_mul (hk : Tendsto k l atTop)
    (hf : SuperpolynomialDecay l k f) : SuperpolynomialDecay l k (k⁻¹ * f) 
goal: Asymptotics.SuperpolynomialDecay l k (k⁻¹ * f)

full_name: Asymptotics.SuperpolynomialDecay.param_inv_mul
code: theorem SuperpolynomialDecay.param_inv_mul (hk : Tendsto k l atTop)
    (hf : SuperpolynomialDecay l k f) : SuperpolynomialDecay l k (f * k⁻¹) 
goal: Asymptotics.SuperpolynomialDecay l k (f * k⁻¹)

full_name: Asymptotics.superpolynomialDecay_param_mul_iff
code: theorem superpolynomialDecay_param_mul_iff (hk : Tendsto k l atTop) :
    SuperpolynomialDecay l k (k * f) ↔ SuperpolynomialDecay l k f 
goal: Asymptotics.SuperpolynomialDecay l k (k * f) ↔ Asymptotics.SuperpolynomialDecay l k f

full_name: Asymptotics.superpolynomialDecay_mul_param_iff
code: theorem superpolynomialDecay_mul_param_iff (hk : Tendsto k l atTop) :
    SuperpolynomialDecay l k (f * k) ↔ SuperpolynomialDecay l k f 
goal: Asymptotics.SuperpolynomialDecay l k (f * k) ↔ Asymptotics.SuperpolynomialDecay l k f

full_name: Asymptotics.superpolynomialDecay_param_pow_mul_iff
code: theorem superpolynomialDecay_param_pow_mul_iff (hk : Tendsto k l atTop) (n : ℕ) :
    SuperpolynomialDecay l k (k ^ n * f) ↔ SuperpolynomialDecay l k f 
goal: Asymptotics.SuperpolynomialDecay l k (k ^ n * f) ↔ Asymptotics.SuperpolynomialDecay l k f

full_name: Asymptotics.superpolynomialDecay_mul_param_pow_iff
code: theorem superpolynomialDecay_mul_param_pow_iff (hk : Tendsto k l atTop) (n : ℕ) :
    SuperpolynomialDecay l k (f * k ^ n) ↔ SuperpolynomialDecay l k f 
goal: Asymptotics.SuperpolynomialDecay l k (f * k ^ n) ↔ Asymptotics.SuperpolynomialDecay l k f

full_name: Asymptotics.superpolynomialDecay_iff_norm_tendsto_zero
code: theorem superpolynomialDecay_iff_norm_tendsto_zero :
    SuperpolynomialDecay l k f ↔ ∀ n : ℕ, Tendsto (fun a : α => ‖k a ^ n * f a‖) l (𝓝 0) 
goal: Asymptotics.SuperpolynomialDecay l k f ↔ ∀ (n : ℕ), Filter.Tendsto (fun a => ‖k a ^ n * f a‖) l (nhds 0)

full_name: Asymptotics.superpolynomialDecay_iff_superpolynomialDecay_norm
code: theorem superpolynomialDecay_iff_superpolynomialDecay_norm :
    SuperpolynomialDecay l k f ↔ SuperpolynomialDecay l (fun a => ‖k a‖) fun a => ‖f a‖ 
goal: Asymptotics.SuperpolynomialDecay l k f ↔ Asymptotics.SuperpolynomialDecay l (fun a => ‖k a‖) fun a => ‖f a‖

full_name: Asymptotics.superpolynomialDecay_iff_isBigO
code: theorem superpolynomialDecay_iff_isBigO (hk : Tendsto k l atTop) :
    SuperpolynomialDecay l k f ↔ ∀ z : ℤ, f =O[l] fun a : α => k a ^ z 
goal: Asymptotics.SuperpolynomialDecay l k f ↔ ∀ (z : ℤ), f =O[l] fun a => k a ^ z

full_name: Asymptotics.superpolynomialDecay_iff_isLittleO
code: theorem superpolynomialDecay_iff_isLittleO (hk : Tendsto k l atTop) :
    SuperpolynomialDecay l k f ↔ ∀ z : ℤ, f =o[l] fun a : α => k a ^ z 
goal: Asymptotics.SuperpolynomialDecay l k f ↔ ∀ (z : ℤ), f =o[l] fun a => k a ^ z