mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Group.Quotient

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝⁸ : SeminormedAddCommGroup M
inst✝⁷ : SeminormedAddCommGroup N
R : Type u_3
inst✝⁶ : Ring R
inst✝⁵ : Module R M
S T : Submodule R M
𝕜 : Type u_4
inst✝⁴ : SeminormedCommRing 𝕜
inst✝³ : Module 𝕜 M
inst✝² : IsBoundedSMul 𝕜 M
inst✝¹ : SMul 𝕜 R
inst✝ : IsScalarTower 𝕜 R M
k : 𝕜
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ : 0 < δ
a : M
h : ∀ (x : ℝ), x - ‖mk a‖ < δ ∧ ‖mk a‖ - x < δ → ‖k‖ * x - ‖k‖ * ‖mk a‖ < ε ∧ ‖k‖ * ‖mk a‖ - ‖k‖ * x < ε
ha : ‖a‖ < ‖mk a‖ + δ
⊢ ‖mk a‖ - ‖a‖ < δ

Tactic:
by

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

Goal state before:
pp: x : ℝ
n : ℕ
⊢ x.sqrtTwoAddSeries (n + 1 + 1) = (√(2 + x)).sqrtTwoAddSeries (n + 1)

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝⁸ : SeminormedAddCommGroup M
inst✝⁷ : SeminormedAddCommGroup N
R : Type u_3
inst✝⁶ : Ring R
inst✝⁵ : Module R M
S T : Submodule R M
𝕜 : Type u_4
inst✝⁴ : SeminormedCommRing 𝕜
inst✝³ : Module 𝕜 M
inst✝² : IsBoundedSMul 𝕜 M
inst✝¹ : SMul 𝕜 R
inst✝ : IsScalarTower 𝕜 R M
k : 𝕜
x : M ⧸ S
ε : ℝ
hε : 0 < ε
⊢ ‖k • x‖ ≤ ‖k‖ * ‖x‖ + ε

Tactic:
by

Goal state after:
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
⊢ s * closedBall 1 δ = cthickening δ s

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝³ : SeminormedAddCommGroup M
inst✝² : SeminormedAddCommGroup N
R : Type u_3
inst✝¹ : Ring R
inst✝ : Module R M
S T : Submodule R M
h : S = T
⊢ ∀ (x : M ⧸ S), ‖(S.quotEquivOfEq T h) x‖ = ‖x‖

Tactic:
by

Goal state after:
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
⊢ s / closedBall 1 δ = cthickening δ s

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝³ : SeminormedAddCommGroup M
inst✝² : SeminormedAddCommGroup N
R : Type u_3
inst✝¹ : Ring R
inst✝ : Module R M
S T : Submodule R M
⊢ map S.mkQ T = map S.mkQ (S ⊔ T)

Tactic:
by

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

Goal state before:
pp: x y : ℝ
h : x ≤ y
n : ℕ
⊢ x.sqrtTwoAddSeries (n + 1) ≤ y.sqrtTwoAddSeries (n + 1)

Tactic:
by

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

Goal state before:
pp: ⊢ cos (π / 2 ^ (0 + 1)) = sqrtTwoAddSeries 0 0 / 2

Tactic:
by

Goal state after:
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:
by

Goal state after:
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⁻¹

Tactic:
by

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

Goal state before:
pp: n : ℕ
A : 1 < 2 ^ (n + 1)
B : π / 2 ^ (n + 1) < π
⊢ 0 < π / 2 ^ (n + 1)

Tactic:
by

Goal state after:
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
x : E
⊢ s * closedBall x δ = x • cthickening δ s

Tactic:
by

Goal state after:
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
x : E
⊢ s / closedBall x δ = x⁻¹ • cthickening δ s

Tactic:
by

Goal state after:
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
x : E
⊢ closedBall x δ * s = x • cthickening δ s

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : SeminormedAddCommGroup M
inst✝¹ : SeminormedAddCommGroup N
R : Type u_3
inst✝ : SeminormedCommRing R
I : Ideal R
ε : ℝ
hε : 0 < ε
ε₁ ε₂ : ℝ
h₁ : 0 < ε₁
h₂ : 0 < ε₂
a : R
ha : ‖a‖ < ‖(mk I) a‖ + ε₁
b : R
hb : ‖b‖ < ‖(mk I) b‖ + ε₂
h :
  ∀ (a_1 b_1 : ℝ),
    a_1 - ‖(mk I) a‖ < ε₁ ∧ ‖(mk I) a‖ - a_1 < ε₁ →
      b_1 - ‖(mk I) b‖ < ε₂ ∧ ‖(mk I) b‖ - b_1 < ε₂ →
        a_1 * b_1 - ‖(mk I) a‖ * ‖(mk I) b‖ < ε ∧ ‖(mk I) a‖ * ‖(mk I) b‖ - a_1 * b_1 < ε
⊢ ‖a‖ - ‖(mk I) a‖ < ε₁

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
⊢ ‖x‖⁻¹ • x ∈ closedBall 0 1

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t : ℝ
ht : 0 ≤ t
x : E
⊢ ‖t • x‖ = t * ‖x‖

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
x y : E
h : r ∈ Icc 0 1
⊢ dist (r • x + (1 - r) • y) x = ‖1 - r‖ * ‖x - y‖

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : SeminormedAddCommGroup M
inst✝¹ : SeminormedAddCommGroup N
R : Type u_3
inst✝ : SeminormedCommRing R
I : Ideal R
ε : ℝ
hε : 0 < ε
ε₁ ε₂ : ℝ
h₁ : 0 < ε₁
h₂ : 0 < ε₂
a : R
ha : ‖a‖ < ‖(mk I) a‖ + ε₁
b : R
hb : ‖b‖ < ‖(mk I) b‖ + ε₂
h :
  ∀ (a_1 b_1 : ℝ),
    a_1 - ‖(mk I) a‖ < ε₁ ∧ ‖(mk I) a‖ - a_1 < ε₁ →
      b_1 - ‖(mk I) b‖ < ε₂ ∧ ‖(mk I) b‖ - b_1 < ε₂ →
        a_1 * b_1 - ‖(mk I) a‖ * ‖(mk I) b‖ < ε ∧ ‖(mk I) a‖ * ‖(mk I) b‖ - a_1 * b_1 < ε
⊢ ‖(mk I) a‖ - ‖a‖ < ε₁

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
x y : E
h : r ∈ Icc 0 1
⊢ ‖1 - r‖ * ‖x - y‖ = (1 - r) * dist y x

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
x y : E
h : r ∈ Icc 0 1
⊢ (1 - r) * dist y x ≤ (1 - 0) * dist y x

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
x y : E
h : r ∈ Icc 0 1
⊢ (1 - 0) * dist y x = dist y x

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : SeminormedAddCommGroup M
inst✝¹ : SeminormedAddCommGroup N
R : Type u_3
inst✝ : SeminormedCommRing R
I : Ideal R
ε : ℝ
hε : 0 < ε
ε₁ ε₂ : ℝ
h₁ : 0 < ε₁
h₂ : 0 < ε₂
a : R
ha : ‖a‖ < ‖(mk I) a‖ + ε₁
b : R
hb : ‖b‖ < ‖(mk I) b‖ + ε₂
h :
  ∀ (a_1 b_1 : ℝ),
    a_1 - ‖(mk I) a‖ < ε₁ ∧ ‖(mk I) a‖ - a_1 < ε₁ →
      b_1 - ‖(mk I) b‖ < ε₂ ∧ ‖(mk I) b‖ - b_1 < ε₂ →
        a_1 * b_1 - ‖(mk I) a‖ * ‖(mk I) b‖ < ε ∧ ‖(mk I) a‖ * ‖(mk I) b‖ - a_1 * b_1 < ε
⊢ ‖b‖ - ‖(mk I) b‖ < ε₂

Tactic:
by

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

Goal state before:
pp: 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
⊢ ContinuousWithinAt (fun c ↦ c • (y - x) + x) (Ico 0 1) 1

Tactic:
by

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

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : SeminormedAddCommGroup M
inst✝¹ : SeminormedAddCommGroup N
R : Type u_3
inst✝ : SeminormedCommRing R
I : Ideal R
ε : ℝ
hε : 0 < ε
ε₁ ε₂ : ℝ
h₁ : 0 < ε₁
h₂ : 0 < ε₂
a : R
ha : ‖a‖ < ‖(mk I) a‖ + ε₁
b : R
hb : ‖b‖ < ‖(mk I) b‖ + ε₂
h :
  ∀ (a_1 b_1 : ℝ),
    a_1 - ‖(mk I) a‖ < ε₁ ∧ ‖(mk I) a‖ - a_1 < ε₁ →
      b_1 - ‖(mk I) b‖ < ε₂ ∧ ‖(mk I) b‖ - b_1 < ε₂ →
        a_1 * b_1 - ‖(mk I) a‖ * ‖(mk I) b‖ < ε ∧ ‖(mk I) a‖ * ‖(mk I) b‖ - a_1 * b_1 < ε
⊢ ‖(mk I) b‖ - ‖b‖ < ε₂

Tactic:
by

Goal state after:
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
a : ℝ
ha : a < R
H : ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n
⊢ ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x

Tactic:
by

Goal state after:
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:
rintro c ⟨hc0, hc1⟩

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
c : ℝ
hc0 : 0 ≤ c
hc1 : c < 1
⊢ (fun c ↦ c • (y - x) + x) c ∈ ball x r
Module: Mathlib.Analysis.Normed.Group.Quotient

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : SeminormedAddCommGroup M
inst✝¹ : SeminormedAddCommGroup N
R : Type u_3
inst✝ : SeminormedCommRing R
I : Ideal R
x y : R ⧸ I
ε : ℝ
hε : 0 < ε
⊢ ‖x * y‖ ≤ ‖x‖ * ‖y‖ + ε

Tactic:
by

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

Goal state before:
pp: 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
⊢ ediam (c • s) = ‖c‖₊ • ediam s

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
⊢ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball x r) = closedBall x r

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
⊢ frontier (ball x r) = sphere x r

Tactic:
rw [frontier, closure_ball x hr, isOpen_ball.interior_eq, closedBall_sdiff_ball]

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
⊢ frontier (ball x r) = sphere x r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: f : ℕ → ℝ
R : ℝ
⊢ [∃ 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:
by

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

Goal state before:
pp: 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
x : Set E
⊢ diam (c • x) = ‖c‖ * diam x

Tactic:
by

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

Goal state before:
pp: n : ℕ
⊢ cos (π / 2 ^ (n + 1 + 1)) = sqrtTwoAddSeries 0 (n + 1) / 2

Tactic:
by

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

Goal state before:
pp: 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))
f : ℝ → E := fun c ↦ c • (y - x) + x
this : f ⁻¹' closedBall x (dist y x) ⊆ Icc (-1) 1
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f

Tactic:
by

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

Goal state before:
pp: 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))
f : ℝ → E := fun c ↦ c • (y - x) + x
this : f ⁻¹' closedBall x (dist y x) ⊆ Icc (-1) 1
⊢ 1 ∈ f ⁻¹' interior (closedBall x (dist y x))

Tactic:
by

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

Goal state before:
pp: n : ℕ
⊢ sin (π / 2 ^ (n + 1)) ^ 2 = 1 - (sqrtTwoAddSeries 0 n / 2) ^ 2

Tactic:
by

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

Goal state before:
pp: 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))
f : ℝ → E := fun c ↦ c • (y - x) + x
this : f ⁻¹' closedBall x (dist y x) ⊆ Icc (-1) 1
⊢ y ∈ ball x (dist y x)

Tactic:
by

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

Goal state before:
pp: n : ℕ
⊢ sin (π / 2 ^ (n + 2)) ^ 2 = 1 / 2 - sqrtTwoAddSeries 0 n / 4

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: R : Type u_2
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
⊢ (fun n ↦ ↑n ^ k) =o[atTop] fun n ↦ r ^ n

Tactic:
by

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

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

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: R : Type u_2
inst✝ : NormedRing R
r : ℝ
hr : 1 < r
⊢ Nat.cast =o[atTop] fun n ↦ r ^ n

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
⊢ frontier (closedBall x r) = sphere x r

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
⊢ interior (sphere x r) = ∅

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
⊢ frontier (sphere x r) = sphere x r

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NontrivialTopology E
c : ℝ
hc : 0 ≤ c
⊢ ∃ x, ‖x‖ = c

Tactic:
by

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

Goal state before:
pp: 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
hc : c ≠ 0
s : Set E
x : E
this : Function.Surjective fun x ↦ c • x
⊢ ↑‖c‖₊ ≠ 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NontrivialTopology E
x : E
r : ℝ
hr : 0 ≤ r
y : E
hy : ‖y‖ = r
⊢ x + y ∈ sphere x r

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NontrivialTopology E
x : E
r : ℝ
⊢ (sphere x r).Nonempty ↔ 0 ≤ r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: R : Type u_2
inst✝ : NormedRing R
k : ℕ
r₁ : R
r₂ : ℝ
h : ‖r₁‖ < r₂
h0 : 0 < ‖r₁‖
A : (fun n ↦ ↑n ^ k) =o[atTop] fun n ↦ (r₂ / ‖r₁‖) ^ n
this : (fun n ↦ r₁ ^ n) =O[atTop] fun n ↦ ‖r₁‖ ^ n
⊢ (fun n ↦ ↑n ^ k * r₁ ^ n) =o[atTop] fun n ↦ r₂ ^ n

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ interior (closedBall x r) = ball x r

Tactic:
by

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

Goal state before:
pp: n : ℕ
⊢ sin (π / 2 ^ (n + 2)) = √(2 - sqrtTwoAddSeries 0 n) / 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: R : Type u_2
inst✝ : NormedRing R
k : ℕ
r₁ : R
r₂ : ℝ
h : ‖r₁‖ < r₂
⊢ (fun n ↦ ↑n ^ k * r₁ ^ n) =o[atTop] fun n ↦ r₂ ^ n

Tactic:
by

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

Goal state before:
pp: 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
hc : c ≠ 0
s : Set E
x : E
⊢ infEDist (c • x) (c • s) = ‖c‖₊ • infEDist x s

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ frontier (closedBall x r) = sphere x r

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ interior (sphere x r) = ∅

Tactic:
by

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

Goal state before:
pp: ⊢ cos (π / 4) = √2 / 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: k : ℕ
r : ℝ
hr : |r| < 1
h0 : r = 0
n : ℕ
hn : 1 ≤ n
⊢ n ∈ {x | (fun x ↦ 0 = (fun n ↦ ↑n ^ k * r ^ n) x) x}

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ frontier (sphere x r) = sphere x r

Tactic:
by

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

Goal state before:
pp: ⊢ sin (π / 4) = √2 / 2

Tactic:
by

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

Goal state before:
pp: 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
hc : c ≠ 0
s : Set E
x : E
⊢ infDist (c • x) (c • s) = ‖c‖ * infDist x s

Tactic:
by

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

Goal state before:
pp: ⊢ cos (π / 8) = √(2 + √2) / 2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Hom.ContinuousEvalConst

Goal state before:
pp: F : Type u_1
α : Type u_2
X : Type u_3
inst✝⁵ : FunLike F α X
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalSpace X
inst✝² : ContinuousEvalConst F α X
F' : Type u_5
inst✝¹ : FunLike F' α X
inst✝ : TopologicalSpace F'
f : F' → F
hc : Continuous[inst✝, inst✝⁴] f
hf : ∀ (g : F'), ⇑(f g) = ⇑g
x : α
⊢ Continuous[inst✝, inst✝³] fun f ↦ f x

Tactic:
by

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

Goal state before:
pp: ⊢ sin (π / 8) = √(2 - √2) / 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: k : ℕ
r : ℝ
hr : |r| < 1
⊢ Tendsto (fun n ↦ ↑n ^ k * r ^ n) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: r : ℝ
k : ℕ
hk : k ≠ 0
⊢ Tendsto (fun n ↦ r / ↑n ^ k) atTop (𝓝 0)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : c ≠ 0
x : E
r : ℝ
⊢ c • ball x r = ball (c • x) (‖c‖ * r)

Tactic:
by

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

Goal state before:
pp: ⊢ cos (π / 16) = √(2 + √(2 + √2)) / 2

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : c ≠ 0
⊢ c • ball 0 1 = ball 0 ‖c‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: r : ℝ
hr : |r| < 1
⊢ Tendsto (fun n ↦ ↑n * r ^ n) atTop (𝓝 0)

Tactic:
by

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

Goal state before:
pp: ⊢ sin (π / 16) = √(2 - √(2 + √2)) / 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: r : ℝ
hr : 0 ≤ r
h'r : r < 1
⊢ Tendsto (fun n ↦ ↑n * r ^ n) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: R : Type u_2
inst✝ : SeminormedRing R
x : R
h : ‖x‖ < 1
⊢ Tendsto (fun n ↦ x ^ n) atTop (𝓝 0)

Tactic:
by

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

Goal state before:
pp: ⊢ cos (π / 32) = √(2 + √(2 + √(2 + √2))) / 2

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : c ≠ 0
x : E
r : ℝ
⊢ c • sphere x r = sphere (c • x) (‖c‖ * r)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : c ≠ 0
x : E
r : ℝ
⊢ c • closedBall x r = closedBall (c • x) (‖c‖ * r)

Tactic:
by

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

Goal state before:
pp: ⊢ sin (π / 32) = √(2 - √(2 + √(2 + √2))) / 2

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set 𝕜
hs : 0 ∉ s
r : ℝ
c : 𝕜
hc : c ∈ s
⊢ c • sphere 0 r = sphere 0 (‖c‖ * r)

Tactic:
by

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

Goal state before:
pp: ⊢ cos (3 * (π / 3)) = cos π

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set 𝕜
hs : 0 ∉ s
r : ℝ
⊢ ⋃ c ∈ s, sphere 0 (‖c‖ * r) = (fun x ↦ ‖x‖) ⁻¹' (fun x ↦ ‖x‖ * r) '' s

Tactic:
by

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

Goal state before:
pp: ⊢ (2 * cos (π / 3) - 1) ^ 2 * (2 * cos (π / 3) + 2) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: R : Type u_2
inst✝¹ : SeminormedRing R
inst✝ : NormMulClass R
x : R
⊢ Tendsto (fun n ↦ x ^ n) atTop (𝓝 0) ↔ ‖x‖ < 1

Tactic:
by

Goal state after: