mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Hom.ContinuousEval

Goal state before:
pp: F : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁶ : FunLike F X Y
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : ContinuousEval F X Y
F' : Type u_5
inst✝¹ : FunLike F' X Y
inst✝ : TopologicalSpace F'
f : F' → F
hc : Continuous[inst✝, inst✝⁵] f
hf : ∀ (g : F'), ⇑(f g) = ⇑g
⊢ Continuous[instTopologicalSpaceProd, inst✝³] fun fx ↦ fx.1 fx.2

Tactic:
by

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

Goal state before:
pp: h₁ : (2 * cos (π / 3) - 1) ^ 2 * (2 * cos (π / 3) + 2) = 0
h : 2 * cos (π / 3) + 2 = 0
⊢ cos π < cos (π / 3)

Tactic:
by

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

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝⁵ : Field R
inst✝⁴ : Field S
inst✝³ : LinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : IsStrictOrderedRing S
inst✝ : Archimedean S
_i : OrderTopology S
v : AbsoluteValue R S
a : R
ha : v a < 1
h_add : Tendsto (fun x ↦ 1 + v a ^ x) atTop (𝓝 (1 + 0))
h_sub : Tendsto (fun x ↦ 1 - v a ^ x) atTop (𝓝 (1 - 0))
⊢ Tendsto ?m.137 atTop (𝓝 1)

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
x : E
s : Set E
hs : Bornology.IsBounded s
u : Set E
hu : u ∈ 𝓝 x
ε : ℝ
εpos : 0 < ε
hε : closedBall x ε ⊆ u
R : ℝ
Rpos : 0 < R
hR : s ⊆ closedBall 0 R
this : closedBall 0 (ε / R) ∈ 𝓝 0
r : 𝕜
hr : r ∈ closedBall 0 (ε / R)
y : E
hy : y ∈ (fun x_1 ↦ -x + x_1) ⁻¹' (r • s)
⊢ ∃ z ∈ s, r • z = -x + y

Tactic:
by

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

Goal state before:
pp: ⊢ cos (π / 3) = 1 / 2

Tactic:
by

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

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝⁵ : Field R
inst✝⁴ : Field S
inst✝³ : LinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : IsStrictOrderedRing S
inst✝ : Archimedean S
_i : OrderTopology S
v : AbsoluteValue R S
a : R
ha : v a < 1
h_add : Tendsto (fun x ↦ 1 + v a ^ x) atTop (𝓝 (1 + 0))
h_sub : Tendsto (fun x ↦ 1 - v a ^ x) atTop (𝓝 (1 - 0))
⊢ Tendsto ?m.138 atTop (𝓝 1)

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
x : E
s : Set E
hs : Bornology.IsBounded s
u : Set E
hu : u ∈ 𝓝 x
ε : ℝ
εpos : 0 < ε
hε : closedBall x ε ⊆ u
R : ℝ
Rpos : 0 < R
hR : s ⊆ closedBall 0 R
this : closedBall 0 (ε / R) ∈ 𝓝 0
r : 𝕜
hr : r ∈ closedBall 0 (ε / R)
y : E
hy : y ∈ (fun x_1 ↦ -x + x_1) ⁻¹' (r • s)
z : E
zs : z ∈ s
hz : r • z = -x + y
⊢ ‖r‖ * ‖z‖ ≤ ε / R * 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
x : E
s : Set E
hs : Bornology.IsBounded s
u : Set E
hu : u ∈ 𝓝 x
ε : ℝ
εpos : 0 < ε
hε : closedBall x ε ⊆ u
R : ℝ
Rpos : 0 < R
hR : s ⊆ closedBall 0 R
this : closedBall 0 (ε / R) ∈ 𝓝 0
r : 𝕜
hr : r ∈ closedBall 0 (ε / R)
y : E
hy : y ∈ (fun x_1 ↦ -x + x_1) ⁻¹' (r • s)
z : E
zs : z ∈ s
hz : r • z = -x + y
⊢ ε / R * R = ε

Tactic:
by

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

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝⁵ : Field R
inst✝⁴ : Field S
inst✝³ : LinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : IsStrictOrderedRing S
inst✝ : Archimedean S
_i : OrderTopology S
v : AbsoluteValue R S
a : R
ha : v a < 1
h_add : Tendsto (fun x ↦ 1 + v a ^ x) atTop (𝓝 (1 + 0))
h_sub : Tendsto (fun x ↦ 1 - v a ^ x) atTop (𝓝 (1 - 0))
n : ℕ
⊢ 1 - v a ^ n ≤ v 1 - v (a ^ 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✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
s : Set E
hs : Bornology.IsBounded s
u : Set E
hu : u ∈ 𝓝 x
ε : ℝ
εpos : 0 < ε
hε : closedBall x ε ⊆ u
R : ℝ
Rpos : 0 < R
hR : s ⊆ closedBall 0 R
this : closedBall 0 (ε / R) ∈ 𝓝 0
r : 𝕜
hr : r ∈ closedBall 0 (ε / R)
y : E
hy : y ∈ (fun x_1 ↦ -x + x_1) ⁻¹' (r • s)
z : E
zs : z ∈ s
hz : r • z = -x + y
I : ‖r • z‖ ≤ ε
⊢ y = x + r • z

Tactic:
by

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

Goal state before:
pp: ⊢ cos (π / 6) = √3 / 2

Tactic:
by

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

Goal state before:
pp: ⊢ cos (π / 6) ^ 2 = 3 / 4

Tactic:
by

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

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝⁵ : Field R
inst✝⁴ : Field S
inst✝³ : LinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : IsStrictOrderedRing S
inst✝ : Archimedean S
_i : OrderTopology S
v : AbsoluteValue R S
a : R
ha : v a < 1
h_add : Tendsto (fun x ↦ 1 + v a ^ x) atTop (𝓝 (1 + 0))
h_sub : Tendsto (fun x ↦ 1 - v a ^ x) atTop (𝓝 (1 - 0))
n : ℕ
⊢ v 1 + v (a ^ n) ≤ 1 + v a ^ n

Tactic:
by

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

Goal state before:
pp: ⊢ sin (π / 6) = 1 / 2

Tactic:
by

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

Goal state before:
pp: ⊢ sin (π / 3) ^ 2 = 3 / 4

Tactic:
by

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

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝⁵ : Field R
inst✝⁴ : Field S
inst✝³ : LinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : IsStrictOrderedRing S
inst✝ : Archimedean S
_i : OrderTopology S
v : AbsoluteValue R S
a : R
ha : v a < 1
⊢ Tendsto (fun n ↦ v (1 / (1 + a ^ n))) atTop (𝓝 1)

Tactic:
by

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

Goal state before:
pp: ⊢ sin (π / 3) = √3 / 2

Tactic:
by

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

Goal state before:
pp: θ : ℝ := π / 5
hθ : θ = π / 5
c : ℝ := cos θ
s : ℝ := sin θ
this : 2 * c = 4 * c ^ 2 - 1
⊢ 4 * c ^ 2 - 2 * c - 1 = 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
x : E
s : Set E
hs : Bornology.IsBounded s
u : Set E
hu : u ∈ 𝓝 x
⊢ ∀ᶠ (r : 𝕜) in 𝓝 0, {x} + r • s ⊆ u

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : 0 < r
⊢ r • ball 0 1 = ball 0 r

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b r : ℝ
ha : 0 ≤ a
hr : 0 < r
this : EqOn (fun x ↦ ‖x‖) id (Ioo a b)
⊢ 0 ∉ Ioo a b

Tactic:
by

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

Goal state before:
pp: θ : ℝ := π / 5
hθ : θ = π / 5
c : ℝ := cos θ
s : ℝ := sin θ
n : ℤ
hn : ↑n * π = π / 5
⊢ n * 5 = 1

Tactic:
by

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

Goal state before:
pp: θ : ℝ := π / 5
hθ : θ = π / 5
c : ℝ := cos θ
s : ℝ := sin θ
⊢ s ≠ 0

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b r : ℝ
ha : 0 ≤ a
hr : 0 < r
⊢ Ioo a b • sphere 0 r = ball 0 (b * r) \ closedBall 0 (a * r)

Tactic:
by

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

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝⁵ : Field R
inst✝⁴ : Field S
inst✝³ : LinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : IsStrictOrderedRing S
inst✝ : Archimedean S
_i : OrderTopology S
v : AbsoluteValue R S
a : R
ha : 1 < v a
⊢ Tendsto (fun n ↦ v (1 / (1 + a ^ n))) atTop (𝓝 0)

Tactic:
by

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

Goal state before:
pp: θ : ℝ := π / 5
hθ : θ = π / 5
c : ℝ := cos θ
s : ℝ := sin θ
hs : s ≠ 0
⊢ s * (2 * c) = 2 * s * c

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ ∃ y, dist x y = b * dist x z ∧ dist y z = a * dist x z

Tactic:
by

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

Goal state before:
pp: θ : ℝ := π / 5
hθ : θ = π / 5
c : ℝ := cos θ
s : ℝ := sin θ
hs : s ≠ 0
⊢ 2 * s * c = sin (2 * θ)

Tactic:
by

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

Goal state before:
pp: θ : ℝ := π / 5
hθ : θ = π / 5
c : ℝ := cos θ
s : ℝ := sin θ
hs : s ≠ 0
⊢ sin (2 * θ) = sin (π - 2 * θ)

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ : ℝ
hδ : 0 ≤ δ
hε : 0 ≤ 0
h : dist x z ≤ 0 + δ
⊢ dist x z ≤ δ

Tactic:
by

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

Goal state before:
pp: θ : ℝ := π / 5
hθ : θ = π / 5
c : ℝ := cos θ
s : ℝ := sin θ
hs : s ≠ 0
⊢ sin (π - 2 * θ) = sin (2 * θ + θ)

Tactic:
by

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

Goal state before:
pp: θ : ℝ := π / 5
hθ : θ = π / 5
c : ℝ := cos θ
s : ℝ := sin θ
hs : s ≠ 0
⊢ sin (2 * θ) * c + cos (2 * θ) * s = 2 * s * c * c + cos (2 * θ) * s

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ ε : ℝ
hδ : 0 ≤ δ
hε : 0 ≤ ε
h : dist x z ≤ ε + δ
hε' : 0 < ε
hεδ : 0 < ε + δ
⊢ ε / (ε + δ) + δ / (ε + δ) = 1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
R✝ : Type u_2
S : Type u_3
inst✝⁷ : Field R✝
inst✝⁶ : Field S
inst✝⁵ : LinearOrder S
v w : AbsoluteValue R✝ S
inst✝⁴ : TopologicalSpace S
inst✝³ : IsStrictOrderedRing S
inst✝² : Archimedean S
_i : OrderTopology S
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
⊢ HasSummableGeomSeries R

Tactic:
by

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

Goal state before:
pp: θ : ℝ := π / 5
hθ : θ = π / 5
c : ℝ := cos θ
s : ℝ := sin θ
hs : s ≠ 0
⊢ 2 * s * c * c + cos (2 * θ) * s = 2 * s * c * c + (2 * c ^ 2 - 1) * s

Tactic:
by

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

Goal state before:
pp: θ : ℝ := π / 5
hθ : θ = π / 5
c : ℝ := cos θ
s : ℝ := sin θ
hs : s ≠ 0
⊢ 2 * s * c * c + (2 * c ^ 2 - 1) * s = s * (2 * c * c) + s * (2 * c ^ 2 - 1)

Tactic:
by

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

Goal state before:
pp: θ : ℝ := π / 5
hθ : θ = π / 5
c : ℝ := cos θ
s : ℝ := sin θ
hs : s ≠ 0
⊢ s * (2 * c * c) + s * (2 * c ^ 2 - 1) = s * (4 * c ^ 2 - 1)

Tactic:
by

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

Goal state before:
pp: ⊢ 4 * cos (π / 5) ^ 2 - 2 * cos (π / 5) - 1 = 0

Tactic:
by

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

Goal state before:
pp: ⊢ (4 • X ^ 2 - 2 • X - C 1).IsRoot (cos (π / 5))

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ ε : ℝ
hδ : 0 ≤ δ
hε : 0 ≤ ε
h : dist x z ≤ ε + δ
⊢ ∃ y, dist x y ≤ δ ∧ dist y z ≤ ε

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝ : NormedRing R
x : R
h : ‖x‖ < 1
hx : Summable fun n ↦ x ^ n
⊢ ‖∑' (b : ℕ), (fun n ↦ x ^ (n + 1)) b‖ ≤ (1 - ‖x‖)⁻¹ - 1

Tactic:
by

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

Goal state before:
pp: c : ℝ := cos (π / 5)
⊢ 4 * (c * c) + -2 * c + -1 = 0

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ ε : ℝ
hδ : 0 ≤ δ
hε : 0 < ε
h : dist x z < ε + δ
⊢ ε / (ε + δ) + δ / (ε + δ) = 1

Tactic:
by

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

Goal state before:
pp: c : ℝ := cos (π / 5)
this : 4 * (c * c) + -2 * c + -1 = 0
⊢ discrim 4 (-2) (-1) = 2 * √5 * (2 * √5)

Tactic:
by

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

Goal state before:
pp: c : ℝ := cos (π / 5)
this : 4 * (c * c) + -2 * c + -1 = 0
hd : discrim 4 (-2) (-1) = 2 * √5 * (2 * √5)
⊢ 4 ≠ 0

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝ : NormedRing R
x : R
h : ‖x‖ < 1
hx : ¬Summable fun n ↦ x ^ n
a✝ : Nontrivial R
this : 1 ≤ ‖1‖
⊢ 0 ≤ 1 - ‖x‖

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ ε : ℝ
hδ : 0 ≤ δ
hε : 0 < ε
h : dist x z < ε + δ
⊢ ∃ y, dist x y ≤ δ ∧ dist y z < ε

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 ≤ ε
h : dist x z < ε + δ
y : E
yz : dist z y ≤ ε
xy : dist y x < δ
⊢ dist x y < δ ∧ dist y z ≤ ε

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 ≤ ε
h : dist x z < ε + δ
⊢ ∃ y, dist x y < δ ∧ dist y z ≤ ε

Tactic:
by

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

Goal state before:
pp: c : ℝ := cos (π / 5)
this : 4 * (c * c) + -2 * c + -1 = 0
hd : discrim 4 (-2) (-1) = 2 * √5 * (2 * √5)
h : c = (- -2 - 2 * √5) / (2 * 4)
⊢ π / 5 ∈ Icc (-(π / 2)) (π / 2)

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : dist x z < ε + δ
⊢ ε / (ε + δ) + δ / (ε + δ) = 1

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝ : NormedRing R
x : R
h : ‖x‖ < 1
⊢ ‖∑' (n : ℕ), x ^ n‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹

Tactic:
by

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

Goal state before:
pp: c : ℝ := cos (π / 5)
this : 4 * (c * c) + -2 * c + -1 = 0
hd : discrim 4 (-2) (-1) = 2 * √5 * (2 * √5)
h : c = (- -2 - 2 * √5) / (2 * 4)
⊢ - -2 - 2 * √5 < 0

Tactic:
by

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

Goal state before:
pp: c : ℝ := cos (π / 5)
this : 4 * (c * c) + -2 * c + -1 = 0
hd : discrim 4 (-2) (-1) = 2 * √5 * (2 * √5)
h : c = (- -2 - 2 * √5) / (2 * 4)
⊢ 0 < 2 * 4

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : dist x z < ε + δ
⊢ ∃ y, dist x y < δ ∧ dist y z < ε

Tactic:
by

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

Goal state before:
pp: ⊢ cos (π / 5) = (1 + √5) / 4

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
⊢ ∑' (i : ℕ), x ^ (i + 1) = ∑' (i : ℕ), x ^ i - 1

Tactic:
by

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

Goal state before:
pp: ⊢ √2 / 2 > 0

Tactic:
by

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

Goal state before:
pp: ⊢ tan (π / 4) = 1

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
⊢ x * ∑' (i : ℕ), x ^ i = ∑' (i : ℕ), x ^ (i + 1)

Tactic:
by

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

Goal state before:
pp: ⊢ tan (π / 2) = 0

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
⊢ Disjoint (ball x δ) (ball y ε) ↔ δ + ε ≤ dist x y

Tactic:
by

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

Goal state before:
pp: ⊢ tan (π / 6) = 1 / √3

Tactic:
by

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

Goal state before:
pp: ⊢ tan (π / 3) = √3

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
⊢ (1 + x) * ∑' (i : ℕ), x ^ i = 2 * ∑' (i : ℕ), x ^ i - 1

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
⊢ ∑' (i : ℕ), x ^ i = (1 - x)⁻¹ʳ

Tactic:
by

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

Goal state before:
pp: x : ℝ
h0x : 0 < x
hxp : x < π / 2
⊢ x < π

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
⊢ HasSum (fun i ↦ x ^ i) (1 - x)⁻¹ʳ

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 ≤ ε
⊢ Disjoint (ball x δ) (closedBall y ε) ↔ δ + ε ≤ dist x y

Tactic:
by

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

Goal state before:
pp: K : Type u_4
inst✝ : NormedDivisionRing K
ξ : K
h : ‖ξ‖ < 1
⊢ ξ ≠ 1

Tactic:
by

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

Goal state before:
pp: x : ℝ
h0x : 0 < x
hxp : x < π / 2
⊢ -(π / 2) < x

Tactic:
by

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

Goal state before:
pp: x : ℝ
h0x : 0 < x
hxp : x < π / 2
⊢ 0 < tan x

Tactic:
by

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

Goal state before:
pp: x : ℝ
h0x : 0 ≤ x
hxp✝ : x ≤ π / 2
h✝ : 0 < x
hxp : x = π / 2
⊢ 0 ≤ tan x

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
δ ε : ℝ
hδ : 0 ≤ δ
hε : 0 < ε
⊢ Disjoint (closedBall x δ) (ball y ε) ↔ δ + ε ≤ dist x y

Tactic:
by

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

Goal state before:
pp: x : ℝ
h0x : 0 ≤ x
hxp : x ≤ π / 2
hx0 : 0 = x
x✝ : x < π / 2 ∨ x = π / 2
⊢ 0 ≤ tan x

Tactic:
by

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

Goal state before:
pp: x : ℝ
hx0 : x < 0
hpx : -(π / 2) < x
⊢ 0 < -x

Tactic:
by

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

Goal state before:
pp: K : Type u_4
inst✝ : NormedDivisionRing K
ξ : K
h : ‖ξ‖ < 1
⊢ HasSum (fun n ↦ ξ ^ n) (1 - ξ)⁻¹

Tactic:
by

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

Goal state before:
pp: K : Type u_4
inst✝ : NormedDivisionRing K
ξ : K
⊢ (Summable fun n ↦ ξ ^ n) ↔ ‖ξ‖ < 1

Tactic:
by

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

Goal state before:
pp: x : ℝ
hx0 : x < 0
hpx : -(π / 2) < x
⊢ -x < π / 2

Tactic:
by

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

Goal state before:
pp: x : ℝ
hx0 : x ≤ 0
hpx : -(π / 2) ≤ x
⊢ 0 ≤ -x

Tactic:
by

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

Goal state before:
pp: x : ℝ
hx0 : x ≤ 0
hpx : -(π / 2) ≤ x
⊢ -x ≤ π / 2

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
δ ε : ℝ
hδ : 0 ≤ δ
hε : 0 ≤ ε
⊢ Disjoint (closedBall x δ) (closedBall y ε) ↔ δ + ε < dist x y

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : R
hr : ‖r‖ < 1
u : ℕ → ℕ
hu : (fun n ↦ ↑(u n)) =O[atTop] fun n ↦ ↑(n ^ k)
r' : ℝ
hrr' : ‖‖r‖‖ < r'
h : r' < 1
n : ℕ
hn : ‖r ^ n‖ ≤ ‖r‖ ^ n
⊢ ‖↑(u n)‖ * ‖r ^ n‖ ≤ ↑(u n) * ‖1‖ * ‖r‖ ^ n

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : R
hr : ‖r‖ < 1
u : ℕ → ℕ
hu : (fun n ↦ ↑(u n)) =O[atTop] fun n ↦ ↑(n ^ k)
r' : ℝ
hrr' : ‖‖r‖‖ < r'
h : r' < 1
n : ℕ
hn : ‖r ^ n‖ ≤ ‖r‖ ^ n
this : ‖↑(u n)‖ * ‖r ^ n‖ ≤ ↑(u n) * ‖1‖ * ‖r‖ ^ n
⊢ ↑(u n) * ‖1‖ * ‖r‖ ^ n = ‖1‖ * (↑(u n) * ‖r‖ ^ n)

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : R
hr : ‖r‖ < 1
u : ℕ → ℕ
hu : (fun n ↦ ↑(u n)) =O[atTop] fun n ↦ ↑(n ^ k)
r' : ℝ
hrr' : ‖‖r‖‖ < r'
h : r' < 1
⊢ (fun n ↦ ‖↑(u n) * r ^ n‖) =O[atTop] fun n ↦ ↑(u n) * ‖r‖ ^ n

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : R
hr : ‖r‖ < 1
u : ℕ → ℕ
hu : (fun n ↦ ↑(u n)) =O[atTop] fun n ↦ ↑(n ^ k)
r' : ℝ
hrr' : ‖‖r‖‖ < r'
h : r' < 1
⊢ (fun n ↦ ↑(n ^ k) * ‖r‖ ^ n) =O[atTop] fun n ↦ r' ^ n

Tactic:
by

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

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ : ℝ
hδ : 0 < δ
s : Set E
x : E
hs : ENNReal.ofReal δ ≤ infEDist x s
z : E
hz : z ∈ s
r : NNReal
h : edist x z < ↑r
⊢ 0 < ↑r - δ

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : R
hr : ‖r‖ < 1
u : ℕ → ℕ
hu : (fun n ↦ ↑(u n)) =O[atTop] fun n ↦ ↑(n ^ k)
⊢ Summable fun n ↦ ‖↑(u n) * r ^ n‖

Tactic:
by

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

Goal state before:
pp: x : ℝ
hx : x ∈ Ioo (-(π / 2)) (π / 2)
y : ℝ
hy : y ∈ Ioo (-(π / 2)) (π / 2)
hlt : x < y
⊢ y - x < π

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : R
hr : ‖r‖ < 1
⊢ Summable fun n ↦ ‖↑n ^ k * r ^ n‖

Tactic:
by

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

Goal state before:
pp: ⊢ StrictMonoOn tan (Ioo (-(π / 2)) (π / 2))

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝ : NormedRing R
r : R
hr : ‖r‖ < 1
⊢ Summable fun n ↦ ‖r ^ n‖

Tactic:
by

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

Goal state before:
pp: x y : ℝ
hx₁ : 0 ≤ x
hy₂ : y < π / 2
hxy : x < y
⊢ -(π / 2) < x

Tactic:
by

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

Goal state before:
pp: ⊢ Function.Periodic tan π

Tactic:
by

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

Goal state before:
pp: ⊢ tan π = 0

Tactic:
by

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

Goal state before:
pp: x : ℝ
⊢ tan (π / 2 - x) = (tan x)⁻¹

Tactic:
by

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

Goal state before:
pp: ⊢ Tendsto sin (𝓝[<] (π / 2)) (𝓝 1)

Tactic:
by

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

Goal state before:
pp: ⊢ Tendsto cos (𝓝[<] (π / 2)) (𝓝[>] 0)

Tactic:
by

Goal state after: