mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: case pos
x : ℝ≥0∞
hx_top : ¬x = ∞
hx_zero : x = 0
⊢ x.log.exp = x

pp: case neg
x : ℝ≥0∞
hx_top : ¬x = ∞
hx_zero : ¬x = 0
⊢ x.log.exp = x

Tactic:
·

Goal state after:
case pos
x : ℝ≥0∞
hx_top : ¬x = ∞
hx_zero : x = 0
⊢ x.log.exp = x
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inl.inr.inr
y : ℝ≥0∞
y_real : 0 < y.toReal
⊢ (∞ * y).log = ⊤ + y.log

Tactic:
·

Goal state after:
case inr.inl.inr.inr
y : ℝ≥0∞
y_real : 0 < y.toReal
⊢ (∞ * y).log = ⊤ + y.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
⊢ (x * y).log = x.log + y.log

Tactic:
·

Goal state after:
case inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
⊢ (x * y).log = x.log + y.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: case e'_5
b : ℝ≥0∞
hb : 1 < b
⊢ ⊤ = ⊤ * b.log

pp: case convert_2
b : ℝ≥0∞
hb : 1 < b
⊢ ⊤ ≠ 0 ∨ b.log ≠ ⊥

pp: case convert_3
b : ℝ≥0∞
hb : 1 < b
⊢ ⊤ ≠ 0 ∨ b.log ≠ ⊤

Tactic:
·

Goal state after:
case e'_5
b : ℝ≥0∞
hb : 1 < b
⊢ ⊤ = ⊤ * b.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr.inl
x : ℝ≥0∞
x_real : 0 < x.toReal
⊢ (x * 0).log = ↑(Real.log x.toReal) + log 0

pp: case inr.inr.inr.inl
x : ℝ≥0∞
x_real : 0 < x.toReal
⊢ (x * ∞).log = ↑(Real.log x.toReal) + ∞.log

pp: case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ (x * y).log = ↑(Real.log x.toReal) + y.log

Tactic:
·

Goal state after:
case inr.inr.inl
x : ℝ≥0∞
x_real : 0 < x.toReal
⊢ (x * 0).log = ↑(Real.log x.toReal) + log 0
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr.inr.inl
x : ℝ≥0∞
x_real : 0 < x.toReal
⊢ (x * ∞).log = ↑(Real.log x.toReal) + ∞.log

pp: case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ (x * y).log = ↑(Real.log x.toReal) + y.log

Tactic:
·

Goal state after:
case inr.inr.inr.inl
x : ℝ≥0∞
x_real : 0 < x.toReal
⊢ (x * ∞).log = ↑(Real.log x.toReal) + ∞.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: case e'_5
b : ℝ≥0∞
hb : b < 1
⊢ ⊥ = ⊤ * b.log

pp: case convert_2
b : ℝ≥0∞
hb : b < 1
⊢ ⊤ ≠ 0 ∨ b.log ≠ ⊥

pp: case convert_3
b : ℝ≥0∞
hb : b < 1
⊢ ⊤ ≠ 0 ∨ b.log ≠ ⊤

Tactic:
·

Goal state after:
case e'_5
b : ℝ≥0∞
hb : b < 1
⊢ ⊥ = ⊤ * b.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ (x * y).log = ↑(Real.log x.toReal) + y.log

Tactic:
·

Goal state after:
case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ (x * y).log = ↑(Real.log x.toReal) + y.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ Real.log (x.toReal * y.toReal) = Real.log x.toReal + Real.log y.toReal

pp: case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ 0 < (x * y).toReal

Tactic:
·

Goal state after:
case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ Real.log (x.toReal * y.toReal) = Real.log x.toReal + Real.log y.toReal
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: case e'_5
b : ℝ≥0∞
hb : 1 < b
⊢ ⊥ = ⊥ * b.log

pp: case convert_2
b : ℝ≥0∞
hb : 1 < b
⊢ ⊥ ≠ 0 ∨ b.log ≠ ⊥

pp: case convert_3
b : ℝ≥0∞
hb : 1 < b
⊢ ⊥ ≠ 0 ∨ b.log ≠ ⊤

Tactic:
·

Goal state after:
case e'_5
b : ℝ≥0∞
hb : 1 < b
⊢ ⊥ = ⊥ * b.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inl
x : ℝ≥0∞
y : ℝ
y_neg : y < 0
⊢ (x ^ y).log = ↑y * x.log

pp: case inr.inl
x : ℝ≥0∞
⊢ (x ^ 0).log = ↑0 * x.log

pp: case inr.inr
x : ℝ≥0∞
y : ℝ
y_pos : 0 < y
⊢ (x ^ y).log = ↑y * x.log

Tactic:
·

Goal state after:
case inl
x : ℝ≥0∞
y : ℝ
y_neg : y < 0
⊢ (x ^ y).log = ↑y * x.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inl.inl
y : ℝ
y_neg : y < 0
⊢ (0 ^ y).log = ↑y * log 0

pp: case inl.inr.inl
y : ℝ
y_neg : y < 0
⊢ (∞ ^ y).log = ↑y * ∞.log

pp: case inl.inr.inr
x : ℝ≥0∞
y : ℝ
y_neg : y < 0
x_real : 0 < x.toReal
⊢ (x ^ y).log = ↑y * x.log

Tactic:
·

Goal state after:
case inl.inl
y : ℝ
y_neg : y < 0
⊢ (0 ^ y).log = ↑y * log 0
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: case e'_5
b : ℝ≥0∞
hb : b < 1
⊢ ⊤ = ⊥ * b.log

pp: case convert_2
b : ℝ≥0∞
hb : b < 1
⊢ ⊥ ≠ 0 ∨ b.log ≠ ⊥

pp: case convert_3
b : ℝ≥0∞
hb : b < 1
⊢ ⊥ ≠ 0 ∨ b.log ≠ ⊤

Tactic:
·

Goal state after:
case e'_5
b : ℝ≥0∞
hb : b < 1
⊢ ⊤ = ⊥ * b.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inl.inr.inl
y : ℝ
y_neg : y < 0
⊢ (∞ ^ y).log = ↑y * ∞.log

pp: case inl.inr.inr
x : ℝ≥0∞
y : ℝ
y_neg : y < 0
x_real : 0 < x.toReal
⊢ (x ^ y).log = ↑y * x.log

Tactic:
·

Goal state after:
case inl.inr.inl
y : ℝ
y_neg : y < 0
⊢ (∞ ^ y).log = ↑y * ∞.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inl.inr.inr
x : ℝ≥0∞
y : ℝ
y_neg : y < 0
x_real : 0 < x.toReal
⊢ (x ^ y).log = ↑y * x.log

Tactic:
·

Goal state after:
case inl.inr.inr
x : ℝ≥0∞
y : ℝ
y_neg : y < 0
x_real : 0 < x.toReal
⊢ (x ^ y).log = ↑y * x.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inl
x : ℝ≥0∞
⊢ (x ^ 0).log = ↑0 * x.log

pp: case inr.inr
x : ℝ≥0∞
y : ℝ
y_pos : 0 < y
⊢ (x ^ y).log = ↑y * x.log

Tactic:
·

Goal state after:
case inr.inl
x : ℝ≥0∞
⊢ (x ^ 0).log = ↑0 * x.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr
x : ℝ≥0∞
y : ℝ
y_pos : 0 < y
⊢ (x ^ y).log = ↑y * x.log

Tactic:
·

Goal state after:
case inr.inr
x : ℝ≥0∞
y : ℝ
y_pos : 0 < y
⊢ (x ^ y).log = ↑y * x.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr.inl
y : ℝ
y_pos : 0 < y
⊢ (0 ^ y).log = ↑y * log 0

pp: case inr.inr.inr.inl
y : ℝ
y_pos : 0 < y
⊢ (∞ ^ y).log = ↑y * ∞.log

pp: case inr.inr.inr.inr
x : ℝ≥0∞
y : ℝ
y_pos : 0 < y
x_real : 0 < x.toReal
⊢ (x ^ y).log = ↑y * x.log

Tactic:
·

Goal state after:
case inr.inr.inl
y : ℝ
y_pos : 0 < y
⊢ (0 ^ y).log = ↑y * log 0
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr.inr.inl
y : ℝ
y_pos : 0 < y
⊢ (∞ ^ y).log = ↑y * ∞.log

pp: case inr.inr.inr.inr
x : ℝ≥0∞
y : ℝ
y_pos : 0 < y
x_real : 0 < x.toReal
⊢ (x ^ y).log = ↑y * x.log

Tactic:
·

Goal state after:
case inr.inr.inr.inl
y : ℝ
y_pos : 0 < y
⊢ (∞ ^ y).log = ↑y * ∞.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr.inr.inr
x : ℝ≥0∞
y : ℝ
y_pos : 0 < y
x_real : 0 < x.toReal
⊢ (x ^ y).log = ↑y * x.log

Tactic:
·

Goal state after:
case inr.inr.inr.inr
x : ℝ≥0∞
y : ℝ
y_pos : 0 < y
x_real : 0 < x.toReal
⊢ (x ^ y).log = ↑y * x.log
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inl
u v : ℕ → EReal
hb : ⊥ ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + ⊥
⊢ (linearGrowthInf fun x ↦ v x + ⊥) ≤ linearGrowthInf v

pp: case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ (linearGrowthInf fun x ↦ v x + b) ≤ linearGrowthInf v

Tactic:
·

Goal state after:
case inl
u v : ℕ → EReal
hb : ⊥ ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + ⊥
⊢ (linearGrowthInf fun x ↦ v x + ⊥) ≤ linearGrowthInf v
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ (linearGrowthInf fun x ↦ v x + b) ≤ linearGrowthInf v

Tactic:
·

Goal state after:
case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ (linearGrowthInf fun x ↦ v x + b) ≤ linearGrowthInf v
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v + 0 = linearGrowthInf v

pp: u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v ≠ ⊥ ∨ 0 ≠ ⊤

pp: u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v ≠ ⊤ ∨ 0 ≠ ⊥

Tactic:
·

Goal state after:
case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v + 0 = linearGrowthInf v
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v ≠ ⊥ ∨ 0 ≠ ⊤

pp: u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v ≠ ⊤ ∨ 0 ≠ ⊥

Tactic:
·

Goal state after:
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v ≠ ⊥ ∨ 0 ≠ ⊤
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v ≠ ⊤ ∨ 0 ≠ ⊥

Tactic:
·

Goal state after:
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v ≠ ⊤ ∨ 0 ≠ ⊥
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inl
u v : ℕ → EReal
hb : ⊥ ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + ⊥
⊢ (linearGrowthSup fun x ↦ v x + ⊥) ≤ linearGrowthSup v

pp: case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ (linearGrowthSup fun x ↦ v x + b) ≤ linearGrowthSup v

Tactic:
·

Goal state after:
case inl
u v : ℕ → EReal
hb : ⊥ ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + ⊥
⊢ (linearGrowthSup fun x ↦ v x + ⊥) ≤ linearGrowthSup v
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ (linearGrowthSup fun x ↦ v x + b) ≤ linearGrowthSup v

Tactic:
·

Goal state after:
case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ (linearGrowthSup fun x ↦ v x + b) ≤ linearGrowthSup v
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthSup v + 0 = linearGrowthSup v

pp: u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthSup v ≠ ⊥ ∨ 0 ≠ ⊤

pp: u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthSup v ≠ ⊤ ∨ 0 ≠ ⊥

Tactic:
·

Goal state after:
case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthSup v + 0 = linearGrowthSup v
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthSup v ≠ ⊥ ∨ 0 ≠ ⊤

pp: u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthSup v ≠ ⊤ ∨ 0 ≠ ⊥

Tactic:
·

Goal state after:
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthSup v ≠ ⊥ ∨ 0 ≠ ⊤
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthSup v ≠ ⊤ ∨ 0 ≠ ⊥

Tactic:
·

Goal state after:
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthSup v ≠ ⊤ ∨ 0 ≠ ⊥
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : SuperpolynomialDecay l k f
z : ℤ
hz : 0 ≤ z
⊢ Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)

pp: case neg
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : SuperpolynomialDecay l k f
z : ℤ
hz : z < 0
⊢ Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : SuperpolynomialDecay l k f
z : ℤ
hz : 0 ≤ z
⊢ Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : SuperpolynomialDecay l k f
z : ℤ
hz : z < 0
⊢ Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : SuperpolynomialDecay l k f
z : ℤ
hz : z < 0
⊢ Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
h : ∀ (z : ℤ), Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)
z : ℤ
⊢ f =O[l] fun a ↦ k a ^ z

pp: case refine_2
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
h : ∀ (z : ℤ), f =O[l] fun a ↦ k a ^ z
z : ℤ
⊢ Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
h : ∀ (z : ℤ), Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)
z : ℤ
⊢ f =O[l] fun a ↦ k a ^ z
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case refine_1
a b : EReal
hb : 0 ≤ b
h : ⊥ < b
n : ℕ
⊢ ⊥ * ↑n ≤ 0

pp: case refine_2
a b : EReal
hb : 0 ≤ b
h : ⊥ < b
n : ℕ
⊢ 0 ≤ b * ↑n

Tactic:
·

Goal state after:
case refine_1
a b : EReal
hb : 0 ≤ b
h : ⊥ < b
n : ℕ
⊢ ⊥ * ↑n ≤ 0
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case refine_2
a b : EReal
hb : 0 ≤ b
h : ⊥ < b
n : ℕ
⊢ 0 ≤ b * ↑n

Tactic:
·

Goal state after:
case refine_2
a b : EReal
hb : 0 ≤ b
h : ⊥ < b
n : ℕ
⊢ 0 ≤ b * ↑n
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
h : ∀ (z : ℤ), f =O[l] fun a ↦ k a ^ z
z : ℤ
⊢ Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
h : ∀ (z : ℤ), f =O[l] fun a ↦ k a ^ z
z : ℤ
⊢ Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)
Module: Mathlib.MeasureTheory.Covering.VitaliFamily

Goal state before:
pp: case inl
X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
s : Set X
f : X → Set (Set X)
fset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : X → Set (Set X) := fun x ↦ f x ∩ v.setsAt x
x : X
hx : x ∈ s
ε : ℝ
εpos : ε > 0
t : Set X
tf : t ∈ f x
ht : t ⊆ closedBall x (min ε δ)
h't : t ∈ v.setsAt x
⊢ ∃ t ∈ g x, t ⊆ closedBall x ε

pp: case inr
X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
s : Set X
f : X → Set (Set X)
fset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : X → Set (Set X) := fun x ↦ f x ∩ v.setsAt x
x : X
hx : x ∈ s
ε : ℝ
εpos : ε > 0
t : Set X
tf : t ∈ f x
ht : t ⊆ closedBall x (min ε δ)
h't : t ∈ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
⊢ ∃ t ∈ g x, t ⊆ closedBall x ε

Tactic:
·

Goal state after:
case inl
X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
s : Set X
f : X → Set (Set X)
fset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : X → Set (Set X) := fun x ↦ f x ∩ v.setsAt x
x : X
hx : x ∈ s
ε : ℝ
εpos : ε > 0
t : Set X
tf : t ∈ f x
ht : t ⊆ closedBall x (min ε δ)
h't : t ∈ v.setsAt x
⊢ ∃ t ∈ g x, t ⊆ closedBall x ε
Module: Mathlib.MeasureTheory.Covering.VitaliFamily

Goal state before:
pp: case inr
X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
s : Set X
f : X → Set (Set X)
fset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : X → Set (Set X) := fun x ↦ f x ∩ v.setsAt x
x : X
hx : x ∈ s
ε : ℝ
εpos : ε > 0
t : Set X
tf : t ∈ f x
ht : t ⊆ closedBall x (min ε δ)
h't : t ∈ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
⊢ ∃ t ∈ g x, t ⊆ closedBall x ε

Tactic:
·

Goal state after:
case inr
X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
s : Set X
f : X → Set (Set X)
fset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : X → Set (Set X) := fun x ↦ f x ∩ v.setsAt x
x : X
hx : x ∈ s
ε : ℝ
εpos : ε > 0
t : Set X
tf : t ∈ f x
ht : t ⊆ closedBall x (min ε δ)
h't : t ∈ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
⊢ ∃ t ∈ g x, t ⊆ closedBall x ε
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: b : ℝ≥0∞
⊢ (expGrowthInf fun n ↦ b ^ n) = b.log

Tactic:
rw [expGrowthInf, ← liminf_const (f := atTop (α := ℕ)) (log b)]
  refine liminf_congr (eventually_atTop.2 ⟨1, fun n n_1 ↦ ?_⟩)
  rw [EReal.div_eq_iff (natCast_ne_bot n) (natCast_ne_top n)
    (zero_lt_one.trans_le (Nat.one_le_cast.2 n_1)).ne.symm, log_pow, mul_comm]

Goal state after:
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: case inl
u v : ℕ → ℝ≥0∞
hb : 0 ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ 0 * v n
⊢ (expGrowthInf fun x ↦ 0 * v x) ≤ expGrowthInf v

pp: case inr
u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ (expGrowthInf fun x ↦ b * v x) ≤ expGrowthInf v

Tactic:
·

Goal state after:
case inl
u v : ℕ → ℝ≥0∞
hb : 0 ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ 0 * v n
⊢ (expGrowthInf fun x ↦ 0 * v x) ≤ expGrowthInf v
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inl
u : ℕ → EReal
h : Monotone u
m : ℕ
hm : ¬u m = ⊥ m
m_n : ∀ᶠ (n : ℕ) in atTop, u m ≤ u n
hm' : u m = ⊤
⊢ 0 ≤ linearGrowthInf u

pp: case inr
u : ℕ → EReal
h : Monotone u
m : ℕ
hm : ¬u m = ⊥ m
m_n : ∀ᶠ (n : ℕ) in atTop, u m ≤ u n
hm' : u m ≠ ⊤
⊢ 0 ≤ linearGrowthInf u

Tactic:
·

Goal state after:
case inl
u : ℕ → EReal
h : Monotone u
m : ℕ
hm : ¬u m = ⊥ m
m_n : ∀ᶠ (n : ℕ) in atTop, u m ≤ u n
hm' : u m = ⊤
⊢ 0 ≤ linearGrowthInf u
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: case inr
u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ (expGrowthInf fun x ↦ b * v x) ≤ expGrowthInf v

Tactic:
·

Goal state after:
case inr
u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ (expGrowthInf fun x ↦ b * v x) ≤ expGrowthInf v
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inr
u : ℕ → EReal
h : Monotone u
m : ℕ
hm : ¬u m = ⊥ m
m_n : ∀ᶠ (n : ℕ) in atTop, u m ≤ u n
hm' : u m ≠ ⊤
⊢ 0 ≤ linearGrowthInf u

Tactic:
·

Goal state after:
case inr
u : ℕ → EReal
h : Monotone u
m : ℕ
hm : ¬u m = ⊥ m
m_n : ∀ᶠ (n : ℕ) in atTop, u m ≤ u n
hm' : u m ≠ ⊤
⊢ 0 ≤ linearGrowthInf u
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inl
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : Tendsto v atTop atTop
m : ℕ
hum : ¬u m = ⊥ m
um_uvn : ∀ᶠ (n : ℕ) in atTop, u m ≤ (u ∘ v) n
hum' : u m = ⊤
⊢ 0 ≤ linearGrowthInf fun x ↦ u m

pp: case inr
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : Tendsto v atTop atTop
m : ℕ
hum : ¬u m = ⊥ m
um_uvn : ∀ᶠ (n : ℕ) in atTop, u m ≤ (u ∘ v) n
hum' : u m ≠ ⊤
⊢ 0 ≤ linearGrowthInf fun x ↦ u m

Tactic:
·

Goal state after:
case inl
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : Tendsto v atTop atTop
m : ℕ
hum : ¬u m = ⊥ m
um_uvn : ∀ᶠ (n : ℕ) in atTop, u m ≤ (u ∘ v) n
hum' : u m = ⊤
⊢ 0 ≤ linearGrowthInf fun x ↦ u m
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: case inr
u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 + expGrowthInf v = expGrowthInf v

pp: u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 ≠ ⊥ ∨ expGrowthInf v ≠ ⊤

pp: u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 ≠ ⊤ ∨ expGrowthInf v ≠ ⊥

Tactic:
·

Goal state after:
case inr
u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 + expGrowthInf v = expGrowthInf v
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inr
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : Tendsto v atTop atTop
m : ℕ
hum : ¬u m = ⊥ m
um_uvn : ∀ᶠ (n : ℕ) in atTop, u m ≤ (u ∘ v) n
hum' : u m ≠ ⊤
⊢ 0 ≤ linearGrowthInf fun x ↦ u m

Tactic:
·

Goal state after:
case inr
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : Tendsto v atTop atTop
m : ℕ
hum : ¬u m = ⊥ m
um_uvn : ∀ᶠ (n : ℕ) in atTop, u m ≤ (u ∘ v) n
hum' : u m ≠ ⊤
⊢ 0 ≤ linearGrowthInf fun x ↦ u m
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 ≠ ⊥ ∨ expGrowthInf v ≠ ⊤

pp: u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 ≠ ⊤ ∨ expGrowthInf v ≠ ⊥

Tactic:
·

Goal state after:
u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 ≠ ⊥ ∨ expGrowthInf v ≠ ⊤
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case pos
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : u = ⊥
⊢ linearGrowthInf (u ∘ v) ≤ (linearGrowthSup fun n ↦ ↑(v n)) * linearGrowthInf u

pp: case neg
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : ¬u = ⊥
⊢ linearGrowthInf (u ∘ v) ≤ (linearGrowthSup fun n ↦ ↑(v n)) * linearGrowthInf u

Tactic:
·

Goal state after:
case pos
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : u = ⊥
⊢ linearGrowthInf (u ∘ v) ≤ (linearGrowthSup fun n ↦ ↑(v n)) * linearGrowthInf u
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 ≠ ⊤ ∨ expGrowthInf v ≠ ⊥

Tactic:
·

Goal state after:
u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 ≠ ⊤ ∨ expGrowthInf v ≠ ⊥