mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: case inr
x : ℝ
hx : x ≠ 0
⊢ x + 1 ≤ rexp x

Tactic:
·

Goal state after:
case inr
x : ℝ
hx : x ≠ 0
⊢ x + 1 ≤ rexp x
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: case inl
t : ℝ
ht' : t ≤ ↑0
⊢ (1 - t / ↑0) ^ 0 ≤ rexp (-t)

pp: case inr
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ (1 - t / ↑n) ^ n ≤ rexp (-t)

Tactic:
·

Goal state after:
case inl
t : ℝ
ht' : t ≤ ↑0
⊢ (1 - t / ↑0) ^ 0 ≤ rexp (-t)
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: case ha
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ 0 ≤ 1 - t / ↑n

pp: case hab
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ 1 - t / ↑n ≤ rexp (-(t / ↑n))

Tactic:
·

Goal state after:
case ha
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ 0 ≤ 1 - t / ↑n
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: case hab
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ 1 - t / ↑n ≤ rexp (-(t / ↑n))

Tactic:
·

Goal state after:
case hab
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ 1 - t / ↑n ≤ rexp (-(t / ↑n))
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: case e'_3
n : ℕ
⊢ (1 + (↑n)⁻¹) ^ n = (1 - -1 / ↑n) ^ n

pp: case e'_4
n : ℕ
⊢ rexp 1 = rexp (- -1)

Tactic:
·

Goal state after:
case e'_3
n : ℕ
⊢ (1 + (↑n)⁻¹) ^ n = (1 - -1 / ↑n) ^ n
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: case e'_4
n : ℕ
⊢ rexp 1 = rexp (- -1)

Tactic:
·

Goal state after:
case e'_4
n : ℕ
⊢ rexp 1 = rexp (- -1)
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: x c : ℝ
hc : 0 < c
⊢ x ≤ c⁻¹ * rexp (c * x)

Tactic:
le_inv_mul_iff₀ hc

Goal state after:
x c : ℝ
hc : 0 < c
⊢ c * x ≤ rexp (c * x)
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: case inl
hx : 0 ≤ 0
hx' : 0 < 2
⊢ rexp 0 ≤ (2 + 0) / (2 - 0)

pp: case inr
x : ℝ
hx : 0 ≤ x
hx' : x < 2
hx₀ : 0 < x
⊢ rexp x ≤ (2 + x) / (2 - x)

Tactic:
·

Goal state after:
case inl
hx : 0 ≤ 0
hx' : 0 < 2
⊢ rexp 0 ≤ (2 + 0) / (2 - 0)
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: case inr
x : ℝ
hx : 0 ≤ x
hx' : x < 2
hx₀ : 0 < x
⊢ rexp x ≤ (2 + x) / (2 - x)

Tactic:
·

Goal state after:
case inr
x : ℝ
hx : 0 ≤ x
hx' : x < 2
hx₀ : 0 < x
⊢ rexp x ≤ (2 + x) / (2 - x)
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: case neg
E'' : Type u_9
F'' : Type u_10
inst✝¹ : NormedAddCommGroup E''
inst✝ : NormedAddCommGroup F''
f : ℕ → E''
g : ℕ → F''
h✝ : ∀ᶠ (x : ℕ) in atTop, ∀ (y : ℕ), x ≤ y → g y = 0 → f y = 0
hrec✝ :
  ∀ᶠ (n₀ : ℕ) in atTop,
    ∃ C₀, ∀ᶠ (n : ℕ) in atTop, ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ n, ‖f m‖ ≤ C * ‖g m‖) → ‖f n‖ ≤ C * ‖g n‖
n₀ : ℕ
h : ∀ (y : ℕ), n₀ ≤ y → g y = 0 → f y = 0
C₀ : ℝ
hrec : ∀ᶠ (x : ℕ) in atTop, ∀ (y : ℕ), x ≤ y → ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ y, ‖f m‖ ≤ C * ‖g m‖) → ‖f y‖ ≤ C * ‖g y‖
n₁ : ℕ
H₁ : n₀ ≤ n₁
H₂ : ∀ (y : ℕ), n₁ ≤ y → ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ y, ‖f m‖ ≤ C * ‖g m‖) → ‖f y‖ ≤ C * ‖g y‖
ubounds : Set ℝ := {C | ∀ m ∈ Finset.Icc n₀ n₁, ‖f m‖ ≤ C * ‖g m‖}
C₁ : ℝ := (Finset.Icc n₀ n₁).sup' ⋯ fun n ↦ ‖f n‖ / ‖g n‖
C₁_mem : C₁ ∈ ubounds
n : ℕ
h_ind : ∀ m < n, n₁ ≤ m → ‖f m‖ ≤ max C₀ C₁ * ‖g m‖
hn : n₁ ≤ n
m : ℕ
hm : m ∈ Finset.Ico n₀ n
hbase : ¬m < n₁
⊢ ‖f m‖ ≤ max C₀ C₁ * ‖g m‖

Tactic:
·

Goal state after:
case neg
E'' : Type u_9
F'' : Type u_10
inst✝¹ : NormedAddCommGroup E''
inst✝ : NormedAddCommGroup F''
f : ℕ → E''
g : ℕ → F''
h✝ : ∀ᶠ (x : ℕ) in atTop, ∀ (y : ℕ), x ≤ y → g y = 0 → f y = 0
hrec✝ :
  ∀ᶠ (n₀ : ℕ) in atTop,
    ∃ C₀, ∀ᶠ (n : ℕ) in atTop, ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ n, ‖f m‖ ≤ C * ‖g m‖) → ‖f n‖ ≤ C * ‖g n‖
n₀ : ℕ
h : ∀ (y : ℕ), n₀ ≤ y → g y = 0 → f y = 0
C₀ : ℝ
hrec : ∀ᶠ (x : ℕ) in atTop, ∀ (y : ℕ), x ≤ y → ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ y, ‖f m‖ ≤ C * ‖g m‖) → ‖f y‖ ≤ C * ‖g y‖
n₁ : ℕ
H₁ : n₀ ≤ n₁
H₂ : ∀ (y : ℕ), n₁ ≤ y → ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ y, ‖f m‖ ≤ C * ‖g m‖) → ‖f y‖ ≤ C * ‖g y‖
ubounds : Set ℝ := {C | ∀ m ∈ Finset.Icc n₀ n₁, ‖f m‖ ≤ C * ‖g m‖}
C₁ : ℝ := (Finset.Icc n₀ n₁).sup' ⋯ fun n ↦ ‖f n‖ / ‖g n‖
C₁_mem : C₁ ∈ ubounds
n : ℕ
h_ind : ∀ m < n, n₁ ≤ m → ‖f m‖ ≤ max C₀ C₁ * ‖g m‖
hn : n₁ ≤ n
m : ℕ
hm : m ∈ Finset.Ico n₀ n
hbase : ¬m < n₁
⊢ ‖f m‖ ≤ max C₀ C₁ * ‖g m‖
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: case inl
⊢ (fun x ↦ cexp x - ∑ i ∈ Finset.range 0, x ^ i / ↑i !) =O[𝓝 0] fun x ↦ x ^ 0

pp: case inr
n : ℕ
hn : 0 < n
⊢ (fun x ↦ cexp x - ∑ i ∈ Finset.range n, x ^ i / ↑i !) =O[𝓝 0] fun x ↦ x ^ n

Tactic:
·

Goal state after:
case inl
⊢ (fun x ↦ cexp x - ∑ i ∈ Finset.range 0, x ^ i / ↑i !) =O[𝓝 0] fun x ↦ x ^ 0
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: case inr
n : ℕ
hn : 0 < n
⊢ (fun x ↦ cexp x - ∑ i ∈ Finset.range n, x ^ i / ↑i !) =O[𝓝 0] fun x ↦ x ^ n

Tactic:
·

Goal state after:
case inr
n : ℕ
hn : 0 < n
⊢ (fun x ↦ cexp x - ∑ i ∈ Finset.range n, x ^ i / ↑i !) =O[𝓝 0] fun x ↦ x ^ n
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
inst✝¹ : Archimedean α
inst✝ : Nontrivial β
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
⊢ ∀ n ≥ 0, |(1 - abv x ^ n) / (1 - abv x)| ≤ 1 / (1 - abv x)

pp: case refine_2
α : Type u_1
β : Type u_2
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
inst✝¹ : Archimedean α
inst✝ : Nontrivial β
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
⊢ ∀ n ≥ 0, (1 - abv x ^ n) / (1 - abv x) ≤ (1 - abv x ^ n.succ) / (1 - abv x)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
inst✝¹ : Archimedean α
inst✝ : Nontrivial β
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
⊢ ∀ n ≥ 0, |(1 - abv x ^ n) / (1 - abv x)| ≤ 1 / (1 - abv x)
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
inst✝¹ : Archimedean α
inst✝ : Nontrivial β
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ (1 - abv x ^ n) / (1 - abv x) ≤ 1 / (1 - abv x)

pp: case refine_1
α : Type u_1
β : Type u_2
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
inst✝¹ : Archimedean α
inst✝ : Nontrivial β
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ 0 ≤ (1 - abv x ^ n) / (1 - abv x)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
inst✝¹ : Archimedean α
inst✝ : Nontrivial β
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ (1 - abv x ^ n) / (1 - abv x) ≤ 1 / (1 - abv x)
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
inst✝¹ : Archimedean α
inst✝ : Nontrivial β
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
⊢ ∀ n ≥ 0, (1 - abv x ^ n) / (1 - abv x) ≤ (1 - abv x ^ n.succ) / (1 - abv x)

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
inst✝¹ : Archimedean α
inst✝ : Nontrivial β
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
⊢ ∀ n ≥ 0, (1 - abv x ^ n) / (1 - abv x) ≤ (1 - abv x ^ n.succ) / (1 - abv x)
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝⁵ : Field α
inst✝⁴ : LinearOrder α
inst✝³ : IsStrictOrderedRing α
inst✝² : Ring β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : Archimedean α
f : ℕ → β
n m : ℕ
hmn : n.succ ≤ m
hr0 : 0 ≤ 0
hr1 : 0 < 1
h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ 0 * abv (f m)
har1 : |0| < 1
⊢ abv (f m) ≤ abv (f n.succ) * 0⁻¹ ^ n.succ * 0 ^ m

pp: case inr
α : Type u_1
β : Type u_2
inst✝⁵ : Field α
inst✝⁴ : LinearOrder α
inst✝³ : IsStrictOrderedRing α
inst✝² : Ring β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : Archimedean α
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)
har1 : |r| < 1
m : ℕ
hmn : n.succ ≤ m
hr : 0 < r
⊢ abv (f m) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ m

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝⁵ : Field α
inst✝⁴ : LinearOrder α
inst✝³ : IsStrictOrderedRing α
inst✝² : Ring β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : Archimedean α
f : ℕ → β
n m : ℕ
hmn : n.succ ≤ m
hr0 : 0 ≤ 0
hr1 : 0 < 1
h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ 0 * abv (f m)
har1 : |0| < 1
⊢ abv (f m) ≤ abv (f n.succ) * 0⁻¹ ^ n.succ * 0 ^ m
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: case inl
b c : ℝ
hb : 0 < b
⊢ Tendsto (fun x ↦ (b * rexp x + c) / x ^ 0) atTop atTop

pp: case inr
b c : ℝ
n : ℕ
hb : 0 < b
hn : n ≠ 0
⊢ Tendsto (fun x ↦ (b * rexp x + c) / x ^ n) atTop atTop

Tactic:
·

Goal state after:
case inl
b c : ℝ
hb : 0 < b
⊢ Tendsto (fun x ↦ (b * rexp x + c) / x ^ 0) atTop atTop
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 : (∃ a ∈ Set.Ioo (-R) R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x
tfae_2_to_1 : (∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =o[atTop] fun x ↦ a ^ x
tfae_3_to_2 : (∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x ↦ a ^ x
tfae_2_to_4 : (∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo 0 R, f =O[atTop] fun x ↦ a ^ x
tfae_4_to_3 : (∃ a ∈ Set.Ioo 0 R, f =O[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x
tfae_4_to_6 : (∃ a ∈ Set.Ioo 0 R, f =O[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n) →
    ∃ a < R, ∃ C, (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n
a : ℝ
ha : a < R
h₀ : 0 < 0 ∨ 0 < R
H : ∀ (n : ℕ), |f n| ≤ 0 * a ^ n
⊢ f = 0

Tactic:
ext n
        simpa using H n

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: case inl
b c : ℝ
n : ℕ
hb : 0 ≠ b
H : ∀ (d e : ℝ), 0 < d → Tendsto (fun x ↦ x ^ n / (d * rexp x + e)) atTop (𝓝 0)
h : 0 < b
⊢ Tendsto (fun x ↦ x ^ n / (b * rexp x + c)) atTop (𝓝 0)

pp: case inr
b c : ℝ
n : ℕ
hb : 0 ≠ b
H : ∀ (d e : ℝ), 0 < d → Tendsto (fun x ↦ x ^ n / (d * rexp x + e)) atTop (𝓝 0)
h : b < 0
⊢ Tendsto (fun x ↦ x ^ n / (b * rexp x + c)) atTop (𝓝 0)

Tactic:
·

Goal state after:
case inl
b c : ℝ
n : ℕ
hb : 0 ≠ b
H : ∀ (d e : ℝ), 0 < d → Tendsto (fun x ↦ x ^ n / (d * rexp x + e)) atTop (𝓝 0)
h : 0 < b
⊢ Tendsto (fun x ↦ x ^ n / (b * rexp x + c)) atTop (𝓝 0)
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: case inr
b c : ℝ
n : ℕ
hb : 0 ≠ b
H : ∀ (d e : ℝ), 0 < d → Tendsto (fun x ↦ x ^ n / (d * rexp x + e)) atTop (𝓝 0)
h : b < 0
⊢ Tendsto (fun x ↦ x ^ n / (b * rexp x + c)) atTop (𝓝 0)

Tactic:
·

Goal state after:
case inr
b c : ℝ
n : ℕ
hb : 0 ≠ b
H : ∀ (d e : ℝ), 0 < d → Tendsto (fun x ↦ x ^ n / (d * rexp x + e)) atTop (𝓝 0)
h : b < 0
⊢ Tendsto (fun x ↦ x ^ n / (b * rexp x + c)) atTop (𝓝 0)
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: case e'_3
b c : ℝ
n : ℕ
hb : 0 ≠ b
H : ∀ (d e : ℝ), 0 < d → Tendsto (fun x ↦ x ^ n / (d * rexp x + e)) atTop (𝓝 0)
h : b < 0
⊢ (fun x ↦ x ^ n / (b * rexp x + c)) = fun x ↦ -(x ^ n / (-b * rexp x + -c))

pp: case e'_5
b c : ℝ
n : ℕ
hb : 0 ≠ b
H : ∀ (d e : ℝ), 0 < d → Tendsto (fun x ↦ x ^ n / (d * rexp x + e)) atTop (𝓝 0)
h : b < 0
⊢ 𝓝 0 = 𝓝 (-0)

Tactic:
·

Goal state after:
case e'_3
b c : ℝ
n : ℕ
hb : 0 ≠ b
H : ∀ (d e : ℝ), 0 < d → Tendsto (fun x ↦ x ^ n / (d * rexp x + e)) atTop (𝓝 0)
h : b < 0
⊢ (fun x ↦ x ^ n / (b * rexp x + c)) = fun x ↦ -(x ^ n / (-b * rexp x + -c))
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: case e'_5
b c : ℝ
n : ℕ
hb : 0 ≠ b
H : ∀ (d e : ℝ), 0 < d → Tendsto (fun x ↦ x ^ n / (d * rexp x + e)) atTop (𝓝 0)
h : b < 0
⊢ 𝓝 0 = 𝓝 (-0)

Tactic:
·

Goal state after:
case e'_5
b c : ℝ
n : ℕ
hb : 0 ≠ b
H : ∀ (d e : ℝ), 0 < d → Tendsto (fun x ↦ x ^ n / (d * rexp x + e)) atTop (𝓝 0)
h : b < 0
⊢ 𝓝 0 = 𝓝 (-0)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: case inl
x : ℝ
hx : |x| ≤ π
h : x < 0
⊢ |sin x| = sin |x|

pp: case inr
x : ℝ
hx : |x| ≤ π
h : 0 ≤ x
⊢ |sin x| = sin |x|

Tactic:
·

Goal state after:
case inl
x : ℝ
hx : |x| ≤ π
h : x < 0
⊢ |sin x| = sin |x|
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: case inr
x : ℝ
hx : |x| ≤ π
h : 0 ≤ x
⊢ |sin x| = sin |x|

Tactic:
·

Goal state after:
case inr
x : ℝ
hx : |x| ≤ π
h : 0 ≤ x
⊢ |sin x| = sin |x|
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: n : ℤ
h : cos (↑n * (2 * π)) = 1
hx₁ : -n < 1
hx₂ : n < 1
⊢ n ≤ 0

Tactic:
by lia

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: n : ℤ
h : cos (↑n * (2 * π)) = 1
hx₁ : -n < 1
hx₂ : n < 1
⊢ 0 ≤ n

Tactic:
by lia

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Pointwise

Goal state before:
pp: E : Type u_1
inst✝ : SeminormedCommGroup E
δ : ℝ
x : E
⊢ {x} / closedBall 1 δ = closedBall x δ

Tactic:
by
  rw [singleton_div_closedBall, div_one]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Pointwise

Goal state before:
pp: case e'_2
E : Type u_1
inst✝ : SeminormedCommGroup E
δ : ℝ
s : Set E
⊢ s = ⋃ i ∈ s, {i}

pp: case e'_3.f
E : Type u_1
inst✝ : SeminormedCommGroup E
δ : ℝ
s : Set E
x✝¹ : E
x✝ : x✝¹ ∈ s
⊢ ball x✝¹ δ = {x✝¹} * ball 1 δ

Tactic:
·

Goal state after:
case e'_2
E : Type u_1
inst✝ : SeminormedCommGroup E
δ : ℝ
s : Set E
⊢ s = ⋃ i ∈ s, {i}
Module: Mathlib.Analysis.Normed.Group.Quotient

Goal state before:
pp: case h₁
M : Type u_1
N : Type u_2
inst✝¹ : SeminormedAddCommGroup M
inst✝ : SeminormedAddCommGroup N
f : NormedAddGroupHom M N
hquot : f.IsQuotient
m : M
⊢ BddBelow ((fun m_1 ↦ ‖m + m_1‖) '' ↑f.ker)

pp: case h₂
M : Type u_1
N : Type u_2
inst✝¹ : SeminormedAddCommGroup M
inst✝ : SeminormedAddCommGroup N
f : NormedAddGroupHom M N
hquot : f.IsQuotient
m : M
⊢ ‖m‖ ∈ (fun m_1 ↦ ‖m + m_1‖) '' ↑f.ker

Tactic:
·

Goal state after:
case h₁
M : Type u_1
N : Type u_2
inst✝¹ : SeminormedAddCommGroup M
inst✝ : SeminormedAddCommGroup N
f : NormedAddGroupHom M N
hquot : f.IsQuotient
m : M
⊢ BddBelow ((fun m_1 ↦ ‖m + m_1‖) '' ↑f.ker)
Module: Mathlib.Analysis.Normed.Group.Quotient

Goal state before:
pp: case h₂
M : Type u_1
N : Type u_2
inst✝¹ : SeminormedAddCommGroup M
inst✝ : SeminormedAddCommGroup N
f : NormedAddGroupHom M N
hquot : f.IsQuotient
m : M
⊢ ‖m‖ ∈ (fun m_1 ↦ ‖m + m_1‖) '' ↑f.ker

Tactic:
·

Goal state after:
case h₂
M : Type u_1
N : Type u_2
inst✝¹ : SeminormedAddCommGroup M
inst✝ : SeminormedAddCommGroup N
f : NormedAddGroupHom M N
hquot : f.IsQuotient
m : M
⊢ ‖m‖ ∈ (fun m_1 ↦ ‖m + m_1‖) '' ↑f.ker
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: n : ℕ
⊢ sqrtTwoAddSeries 0 n < 2 ^ 2 - 2

pp: n : ℕ
⊢ 0 ≤ 2 + sqrtTwoAddSeries 0 n

Tactic:
·

Goal state after:
n : ℕ
⊢ sqrtTwoAddSeries 0 n < 2 ^ 2 - 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: n : ℕ
⊢ 0 ≤ 2 + sqrtTwoAddSeries 0 n

Tactic:
·

Goal state after:
n : ℕ
⊢ 0 ≤ 2 + sqrtTwoAddSeries 0 n
Module: Mathlib.Analysis.Normed.Group.Pointwise

Goal state before:
pp: E : Type u_1
inst✝ : SeminormedCommGroup E
δ : ℝ
s : Set E
hs : IsCompact s
hδ : 0 ≤ δ
⊢ closedBall 1 δ * s = cthickening δ s

Tactic:
rw [mul_comm, hs.mul_closedBall_one hδ]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.RCLike.Real

Goal state before:
pp: case e'_5
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
y : E
hy : y ∈ closedBall x r
this : ContinuousWithinAt (fun c ↦ c • (y - x) + x) (Ico 0 1) 1
⊢ y = 1 • (y - x) + x

pp: case convert_2
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
y : E
hy : y ∈ closedBall x r
this : ContinuousWithinAt (fun c ↦ c • (y - x) + x) (Ico 0 1) 1
⊢ 1 ∈ closure (Ico 0 1)

pp: case convert_3
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
y : E
hy : y ∈ closedBall x r
this : ContinuousWithinAt (fun c ↦ c • (y - x) + x) (Ico 0 1) 1
⊢ MapsTo (fun c ↦ c • (y - x) + x) (Ico 0 1) (ball x r)

Tactic:
·

Goal state after:
case e'_5
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
y : E
hy : y ∈ closedBall x r
this : ContinuousWithinAt (fun c ↦ c • (y - x) + x) (Ico 0 1) 1
⊢ y = 1 • (y - x) + x
Module: Mathlib.Analysis.Normed.Module.RCLike.Real

Goal state before:
pp: case convert_2
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
y : E
hy : y ∈ closedBall x r
this : ContinuousWithinAt (fun c ↦ c • (y - x) + x) (Ico 0 1) 1
⊢ 1 ∈ closure (Ico 0 1)

pp: case convert_3
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
y : E
hy : y ∈ closedBall x r
this : ContinuousWithinAt (fun c ↦ c • (y - x) + x) (Ico 0 1) 1
⊢ MapsTo (fun c ↦ c • (y - x) + x) (Ico 0 1) (ball x r)

Tactic:
·

Goal state after:
case convert_2
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
y : E
hy : y ∈ closedBall x r
this : ContinuousWithinAt (fun c ↦ c • (y - x) + x) (Ico 0 1) 1
⊢ 1 ∈ closure (Ico 0 1)
Module: Mathlib.Analysis.Normed.Module.RCLike.Real

Goal state before:
pp: case convert_3
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
y : E
hy : y ∈ closedBall x r
this : ContinuousWithinAt (fun c ↦ c • (y - x) + x) (Ico 0 1) 1
⊢ MapsTo (fun c ↦ c • (y - x) + x) (Ico 0 1) (ball x r)

Tactic:
·

Goal state after:
case convert_3
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
y : E
hy : y ∈ closedBall x r
this : ContinuousWithinAt (fun c ↦ c • (y - x) + x) (Ico 0 1) 1
⊢ MapsTo (fun c ↦ c • (y - x) + x) (Ico 0 1) (ball x r)
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : NormSMulClass 𝕜 E
s : Set E
⊢ ‖0‖₊ • ediam s ≤ ediam (0 • s)

pp: case inr
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : NormSMulClass 𝕜 E
c : 𝕜
s : Set E
hc : c ≠ 0
⊢ ‖c‖₊ • ediam s ≤ ediam (c • s)

Tactic:
·

Goal state after:
case inl
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : NormSMulClass 𝕜 E
s : Set E
⊢ ‖0‖₊ • ediam s ≤ ediam (0 • s)
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: case inl.inl
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : NormSMulClass 𝕜 E
⊢ ‖0‖₊ • ediam ∅ ≤ ediam (0 • ∅)

pp: case inl.inr
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : NormSMulClass 𝕜 E
s : Set E
hs : s.Nonempty
⊢ ‖0‖₊ • ediam s ≤ ediam (0 • s)

Tactic:
·

Goal state after:
case inl.inl
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : NormSMulClass 𝕜 E
⊢ ‖0‖₊ • ediam ∅ ≤ ediam (0 • ∅)
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : NormSMulClass 𝕜 E
c : 𝕜
s : Set E
hc : c ≠ 0
⊢ ‖c‖₊ • ediam s ≤ ediam (c • s)

Tactic:
·

Goal state after:
case inr
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : NormSMulClass 𝕜 E
c : 𝕜
s : Set E
hc : c ≠ 0
⊢ ‖c‖₊ • ediam s ≤ ediam (c • s)
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 : (∃ a ∈ Set.Ioo (-R) R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x
tfae_2_to_1 : (∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =o[atTop] fun x ↦ a ^ x
tfae_3_to_2 : (∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x ↦ a ^ x
tfae_2_to_4 : (∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo 0 R, f =O[atTop] fun x ↦ a ^ x
tfae_4_to_3 : (∃ a ∈ Set.Ioo 0 R, f =O[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x
tfae_4_to_6 : (∃ a ∈ Set.Ioo 0 R, f =O[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n) →
    ∃ a < R, ∃ C, (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a < R, ∃ C, (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n) → ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x
tfae_2_to_8 : (∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n
tfae_8_to_7 : (∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n) → ∃ a < R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n
tfae_7_to_3 : (∃ a < R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n) → ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x
⊢ [∃ a ∈ Set.Ioo (-R) R, f =o[atTop] fun x ↦ a ^ x, ∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x ↦ a ^ x,
      ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x, ∃ a ∈ Set.Ioo 0 R, f =O[atTop] fun x ↦ a ^ x,
      ∃ a < R, ∃ C, (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n, ∃ a < R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n].TFAE

Tactic:
tfae_finish

Goal state after:
Module: Mathlib.Analysis.Normed.Module.RCLike.Real

Goal state before:
pp: case inl
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr✝ : r ≠ 0
hr : r < 0
⊢ interior (closedBall x r) = ball x r

pp: case inr
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr✝ : r ≠ 0
hr : 0 < r
⊢ interior (closedBall x r) = ball x r

Tactic:
·

Goal state after:
case inl
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr✝ : r ≠ 0
hr : r < 0
⊢ interior (closedBall x r) = ball x r
Module: Mathlib.Analysis.Normed.Module.RCLike.Real

Goal state before:
pp: case inr.inl
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr✝¹ : r ≠ 0
hr✝ : 0 < r
y : E
hy : y ∈ interior (closedBall x r)
hr : dist y x < r
⊢ y ∈ ball x r

pp: case inr.inr
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
hr✝ : dist y x ≠ 0
hr : 0 < dist y x
hy : y ∈ interior (closedBall x (dist y x))
⊢ y ∈ ball x (dist y x)

Tactic:
·

Goal state after:
case inr.inl
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr✝¹ : r ≠ 0
hr✝ : 0 < r
y : E
hy : y ∈ interior (closedBall x r)
hr : dist y x < r
⊢ y ∈ ball x r