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

Goal state before:
pp: R : Type u_2
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
this : Tendsto (fun x ↦ x ^ k) (𝓝[>] 1) (𝓝 1)
r' : ℝ
hr' : r' ^ k < r
h1 : 1 < r'
h0 : 0 ≤ r'
n : ℕ
⊢ ↑n ≤ (r' - 1)⁻¹ * ‖r' ^ n‖

Tactic:
simpa using

Goal state after:
R : Type u_2
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
this : Tendsto (fun x ↦ x ^ k) (𝓝[>] 1) (𝓝 1)
r' : ℝ
hr' : r' ^ k < r
h1 : 1 < r'
h0 : 0 ≤ r'
n : ℕ
⊢ ↑n ≤ (r' - 1)⁻¹ * r' ^ n
Module: Mathlib.Analysis.Normed.Module.RCLike.Real

Goal state before:
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))
f : ℝ → E := fun c ↦ c • (y - x) + x
c : ℝ
hc : c ∈ f ⁻¹' closedBall x (dist y x)
⊢ ‖c‖ * dist y x ≤ 1 * dist y x

Tactic:
simpa using

Goal state after:
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))
f : ℝ → E := fun c ↦ c • (y - x) + x
c : ℝ
hc : c ∈ f ⁻¹' closedBall x (dist y x)
⊢ |c| * ‖y - x‖ ≤ ‖y - x‖
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:
simpa using

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

Goal state after:
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
⊢ ‖y‖ = r
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:
simpa using

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

Goal state before:
pp: case neg
k : ℕ
r : ℝ
hr : |r| < 1
h0 : ¬r = 0
hr' : 1 < |r|⁻¹
⊢ Tendsto (fun x ↦ ‖↑x ^ k * r ^ x‖) atTop (𝓝 0)

Tactic:
simpa using

Goal state after:
case neg
k : ℕ
r : ℝ
hr : |r| < 1
h0 : ¬r = 0
hr' : 1 < |r|⁻¹
⊢ Tendsto (fun x ↦ ↑x ^ k * |r| ^ x) atTop (𝓝 0)
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:
simpa using

Goal state after:
r : ℝ
k : ℕ
hk : k ≠ 0
⊢ Tendsto (fun n ↦ r / ↑n ^ k) atTop (𝓝 0)
Module: Mathlib.Analysis.SpecificLimits.Normed

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

Tactic:
simpa using

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

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

Goal state after:
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 ↦ (f fx.1) fx.2
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:
simpa using

Goal state after:
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)
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:
simpa using

Goal state after:
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
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:
simpa using

Goal state after:
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)
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
this : y = x + r • z
⊢ y ∈ closedBall x ε

Tactic:
simpa using

Goal state after:
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
this : y = x + r • z
⊢ ‖r • z‖ ≤ ε
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 (a ^ n) - v 1) atTop atTop

Tactic:
simpa using

Goal state after:
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 a ^ n - 1) atTop atTop
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

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

Tactic:
simpa using

Goal state after:
⊢ 4 * cos (π / 5) ^ 2 - 2 * cos (π / 5) - 1 = 0
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
⊢ dist z x < δ + ε

Tactic:
simpa using

Goal state after:
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
⊢ dist x z < δ + ε
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

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

Tactic:
cos_pi_div_four,

Goal state after:
⊢ sin (π / 4) / (√2 / 2) = 1
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: K : Type u_4
inst✝ : NormedDivisionRing K
ξ : K
h : ‖ξ‖ < 1
xi_ne_one : ξ ≠ 1
A : Tendsto (fun n ↦ (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))
⊢ Tendsto (fun n ↦ ∑ i ∈ Finset.range n, ξ ^ i) atTop (𝓝 (1 - ξ)⁻¹)

Tactic:
simpa using

Goal state after:
K : Type u_4
inst✝ : NormedDivisionRing K
ξ : K
h : ‖ξ‖ < 1
xi_ne_one : ξ ≠ 1
A : Tendsto (fun n ↦ (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))
⊢ Tendsto (fun n ↦ (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 (1 - ξ)⁻¹)
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:
simpa using

Goal state after:
R : Type u_4
inst✝ : NormedRing R
r : R
hr : ‖r‖ < 1
⊢ Summable fun n ↦ ‖r ^ n‖
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: case zero
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
r : R
hr : ‖r‖ < 1
⊢ HasSum (fun n ↦ ↑((n + 0).choose 0) * r ^ n) ((1 - r)⁻¹ʳ ^ (0 + 1))

Tactic:
simpa using

Goal state after:
case zero
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
r : R
hr : ‖r‖ < 1
⊢ HasSum (fun n ↦ r ^ n) (1 - r)⁻¹ʳ
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: n : ℕ
⊢ cos (↑n * (2 * ↑π) + ↑π) = -1

Tactic:
simpa using

Goal state after:
n : ℕ
⊢ cos (↑n * (2 * ↑π) + ↑π) = -1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: n : ℤ
⊢ cos (↑n * (2 * ↑π) + ↑π) = -1

Tactic:
simpa using

Goal state after:
n : ℤ
⊢ cos (↑n * (2 * ↑π) + ↑π) = -1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: n : ℕ
⊢ cos (↑n * (2 * ↑π) - ↑π) = -1

Tactic:
simpa using

Goal state after:
n : ℕ
⊢ cos (↑n * (2 * ↑π) - ↑π) = -1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: n : ℤ
⊢ cos (↑n * (2 * ↑π) - ↑π) = -1

Tactic:
simpa using

Goal state after:
n : ℤ
⊢ cos (↑n * (2 * ↑π) - ↑π) = -1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: ⊢ Function.Antiperiodic (fun x ↦ cexp (x * I)) ↑π

Tactic:
simpa using

Goal state after:
⊢ Function.Antiperiodic (fun x ↦ cexp (x * I)) ↑π
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_1
inst✝ : SeminormedAddCommGroup α
C r : ℝ
hr : r < 1
u : ℕ → α
h : ∀ (n : ℕ), ‖u n - u (n + 1)‖ ≤ C * r ^ n
⊢ ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ C * r ^ n

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : SeminormedAddCommGroup α
C r : ℝ
hr : r < 1
u : ℕ → α
h : ∀ (n : ℕ), ‖u n - u (n + 1)‖ ≤ C * r ^ n
⊢ ∀ (n : ℕ), ‖u n - u (n + 1)‖ ≤ C * r ^ n
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_1
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
h : CauchySeq fun n ↦ ∑ k ∈ Finset.range n, f k
b : ℕ → ℝ
left✝ : ∀ (n : ℕ), 0 ≤ b n
key : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (∑ k ∈ Finset.range n, f k) (∑ k ∈ Finset.range m, f k) ≤ b N
right✝ : Tendsto b atTop (𝓝 0)
n : ℕ
⊢ ‖f n‖ ≤ b 0

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
h : CauchySeq fun n ↦ ∑ k ∈ Finset.range n, f k
b : ℕ → ℝ
left✝ : ∀ (n : ℕ), 0 ≤ b n
key : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (∑ k ∈ Finset.range n, f k) (∑ k ∈ Finset.range m, f k) ≤ b N
right✝ : Tendsto b atTop (𝓝 0)
n : ℕ
⊢ ‖f n‖ ≤ b 0
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
r : X
f : C(Y, Z)
K : Set Y
hK : IsCompact K
U : Set (X × Z)
hU : IsOpen[instTopologicalSpaceProd] U
H : MapsTo (⇑((const Y (r, f).1).prodMk (r, f).2)) K U
V : Set X
W : Set Z
hV : IsOpen[inst✝²] V
hW : IsOpen[inst✝] W
hrV : {r} ⊆ V
hKW : ⇑f '' K ⊆ W
hVW : V ×ˢ W ⊆ U
⊢ r ∈ V

Tactic:
simpa using

Goal state after:
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
r : X
f : C(Y, Z)
K : Set Y
hK : IsCompact K
U : Set (X × Z)
hU : IsOpen[instTopologicalSpaceProd] U
H : MapsTo (⇑((const Y (r, f).1).prodMk (r, f).2)) K U
V : Set X
W : Set Z
hV : IsOpen[inst✝²] V
hW : IsOpen[inst✝] W
hrV : {r} ⊆ V
hKW : ⇑f '' K ⊆ W
hVW : V ×ˢ W ⊆ U
⊢ r ∈ V
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
x y : X
F : Set X
γ : Path x y
γ_in : ∀ (t : ↑I), γ t ∈ F
this : γ 0 ∈ F ∧ γ 1 ∈ F
⊢ x ∈ F ∧ y ∈ F

Tactic:
simpa using

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
x y : X
F : Set X
γ : Path x y
γ_in : ∀ (t : ↑I), γ t ∈ F
this : γ 0 ∈ F ∧ γ 1 ∈ F
⊢ x ∈ F ∧ y ∈ F
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y : X
F : Set X
f : X → Y
hf : IsInducing f
hx : x ∈ F
hy : y ∈ F
γ : Path (f x) (f y)
γ' : ↑I → X
hγ'F : ∀ (t : ↑I), γ' t ∈ F
hγ' : ∀ (t : ↑I), f (γ' t) = γ t
h₀ : x ⤳ γ' 0
h₁ : γ' 1 ⤳ y
⊢ Continuous[_, inst✝¹] γ'

Tactic:
simpa using

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y : X
F : Set X
f : X → Y
hf : IsInducing f
hx : x ∈ F
hy : y ∈ F
γ : Path (f x) (f y)
γ' : ↑I → X
hγ'F : ∀ (t : ↑I), γ' t ∈ F
hγ' : ∀ (t : ↑I), f (γ' t) = γ t
h₀ : x ⤳ γ' 0
h₁ : γ' 1 ⤳ y
⊢ Continuous[_, inst✝] fun x_1 ↦ γ x_1
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
G : Type u_4
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
g : G
hg : g ∈ (Submonoid.pathComponentOne G).carrier
⊢ g⁻¹ ∈ (Submonoid.pathComponentOne G).carrier

Tactic:
by simpa using! hg.inv

Goal state after:
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: G : Type u_4
inst✝² : InvolutiveInv G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousInv G
s : Set G
hs : IsPathConnected s
a : G
ha_mem : a ∈ s
ha : ∀ ⦃y : G⦄, y ∈ s → JoinedIn s a y
x : G
hx : x ∈ s⁻¹
⊢ JoinedIn s⁻¹ a⁻¹ x

Tactic:
simpa using

Goal state after:
G : Type u_4
inst✝² : InvolutiveInv G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousInv G
s : Set G
hs : IsPathConnected s
a : G
ha_mem : a ∈ s
ha : ∀ ⦃y : G⦄, y ∈ s → JoinedIn s a y
x : G
hx : x ∈ s⁻¹
⊢ JoinedIn s⁻¹ a⁻¹ x
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f g : C(X, Y)
h : ⇑f ⤳ ⇑g
this : ∀ (K : Set X), IsCompact K → ∀ (U : Set Y), IsOpen[inst✝] U → MapsTo (⇑g) K U → MapsTo (⇑f) K U
⊢ f ⤳ g

Tactic:
by
    simpa [specializes_iff_pure, nhds_compactOpen]

Goal state after:
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
U W : Set X
hW : IsPathConnected W
hWU : W ⊆ U
⊢ IsPathConnected (Subtype.val ⁻¹' W)

Tactic:
IsInducing.subtypeVal.isPathConnected_iff,

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
U W : Set X
hW : IsPathConnected W
hWU : W ⊆ U
⊢ IsPathConnected (Subtype.val '' Subtype.val ⁻¹' W)
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : T2Space Y
ι : Type u_6
l : Filter ι
inst✝ : l.NeBot
F : ι → C(X, Y)
f : (s : Set X) → IsCompact s → C(↑s, Y)
hf : ∀ (s : Set X) (a : IsCompact s), Tendsto (fun i ↦ restrict s (F i)) l (𝓝 (f s a))
s₁ : Set X
hs₁ : IsCompact s₁
s₂ : Set X
hs₂ : IsCompact s₂
x : X
hxs₁ : x ∈ s₁
hxs₂ : x ∈ s₂
⊢ (f s₁ hs₁) ⟨x, hxs₁⟩ = (f s₂ hs₂) ⟨x, hxs₂⟩

Tactic:
have := isCompact_iff_compactSpace.mp hs₁

Goal state after:
X : Type u_2
Y : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : T2Space Y
ι : Type u_6
l : Filter ι
inst✝ : l.NeBot
F : ι → C(X, Y)
f : (s : Set X) → IsCompact s → C(↑s, Y)
hf : ∀ (s : Set X) (a : IsCompact s), Tendsto (fun i ↦ restrict s (F i)) l (𝓝 (f s a))
s₁ : Set X
hs₁ : IsCompact s₁
s₂ : Set X
hs₂ : IsCompact s₂
x : X
hxs₁ : x ∈ s₁
hxs₂ : x ∈ s₂
this : CompactSpace ↑s₁
⊢ (f s₁ hs₁) ⟨x, hxs₁⟩ = (f s₂ hs₂) ⟨x, hxs₂⟩
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
F : Set X
⊢ IsPathConnected F ↔ IsPathConnected univ

Tactic:
IsInducing.subtypeVal.isPathConnected_iff,

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
F : Set X
⊢ IsPathConnected F ↔ IsPathConnected (Subtype.val '' univ)
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x✝ y z : X
ι : Type u_3
F : Set X
inst✝ : PathConnectedSpace X
x : X
_x_in : x ∈ univ
hx : pathComponentIn univ x = univ
⊢ pathComponent x = univ

Tactic:
simpa using

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x✝ y z : X
ι : Type u_3
F : Set X
inst✝ : PathConnectedSpace X
x : X
_x_in : x ∈ univ
hx : pathComponentIn univ x = univ
⊢ pathComponent x = univ
Module: Mathlib.Topology.Instances.AddCircle.Real

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
j : ℕ
⊢ toAddCircle ↑j = ↑(↑j / ↑N)

Tactic:
simpa using

Goal state after:
N : ℕ
inst✝ : NeZero N
j : ℕ
⊢ toAddCircle ↑j = ↑(↑j / ↑N)
Module: Mathlib.Topology.Instances.AddCircle.Real

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
x y : ZMod N
hxy : ↑(↑x.val / ↑N) = ↑(↑y.val / ↑N)
this : 0 < ↑N
⊢ ↑x.val / ↑N < 0 + 1

Tactic:
simpa using

Goal state after:
N : ℕ
inst✝ : NeZero N
x y : ZMod N
hxy : ↑(↑x.val / ↑N) = ↑(↑y.val / ↑N)
this : 0 < ↑N
⊢ x.val < N
Module: Mathlib.Topology.Instances.AddCircle.Real

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
x y : ZMod N
hxy : ↑(↑x.val / ↑N) = ↑(↑y.val / ↑N)
this : 0 < ↑N
⊢ ↑y.val / ↑N < 0 + 1

Tactic:
simpa using

Goal state after:
N : ℕ
inst✝ : NeZero N
x y : ZMod N
hxy : ↑(↑x.val / ↑N) = ↑(↑y.val / ↑N)
this : 0 < ↑N
⊢ y.val < N
Module: Mathlib.Analysis.Normed.Group.AddCircle

Goal state before:
pp: case refine_1.left
x r : ℝ
hr : ∀ (m : ℝ), ↑m = ↑x → r ≤ ‖m‖
⊢ r ≤ fract x

Tactic:
simpa using

Goal state after:
case refine_1.left
x r : ℝ
hr : ∀ (m : ℝ), ↑m = ↑x → r ≤ ‖m‖
⊢ r ≤ fract x
Module: Mathlib.Analysis.Normed.Group.AddCircle

Goal state before:
pp: case refine_1.right
x r : ℝ
hr : ∀ (m : ℝ), ↑m = ↑x → r ≤ ‖m‖
⊢ r ≤ 1 - fract x

Tactic:
simpa using

Goal state after:
case refine_1.right
x r : ℝ
hr : ∀ (m : ℝ), ↑m = ↑x → r ≤ ‖m‖
⊢ r ≤ 1 - fract x
Module: Mathlib.Analysis.Normed.Group.AddCircle

Goal state before:
pp: case refine_2
x : ℝ
⊢ ∀ (m : ℝ), ↑m = ↑x → |x - ↑(round x)| ≤ |m|

Tactic:
simpa using

Goal state after:
case refine_2
x : ℝ
⊢ ∀ (a : ℤ), |x - ↑(round x)| ≤ |x - ↑a|
Module: Mathlib.Analysis.Normed.Group.AddCircle

Goal state before:
pp: p : ℝ
hp : p ≠ 0
x✝ : AddCircle p
ε : ℝ
hε : |p| / 2 ≤ ε
x : AddCircle p
⊢ x ∈ closedBall x✝ ε

Tactic:
simpa using

Goal state after:
p : ℝ
hp : p ≠ 0
x✝ : AddCircle p
ε : ℝ
hε : |p| / 2 ≤ ε
x : AddCircle p
⊢ ‖x - x✝‖ ≤ ε
Module: Mathlib.Topology.Path

Goal state before:
pp: X✝ : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X✝
inst✝¹ : TopologicalSpace Y
x y z : X✝
ι : Type u_3
γ✝ : Path x y
X : Type u_4
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ : ℝ
h₁ : t₀ ≤ ↑0
h₂ : ¬↑0 ≤ t₁
h₄ : t₀ ≤ t₁
⊢ t₁ < 0

Tactic:
simpa using

Goal state after:
X✝ : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X✝
inst✝¹ : TopologicalSpace Y
x y z : X✝
ι : Type u_3
γ✝ : Path x y
X : Type u_4
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ : ℝ
h₁ : t₀ ≤ ↑0
h₂ : ¬↑0 ≤ t₁
h₄ : t₀ ≤ t₁
⊢ t₁ < 0
Module: Mathlib.Topology.Path

Goal state before:
pp: X✝ : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X✝
inst✝¹ : TopologicalSpace Y
x y z : X✝
ι : Type u_3
γ✝ : Path x y
X : Type u_4
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ : ℝ
h₁ : ¬t₀ ≤ ↑1
h₃ : t₀ ≤ t₁
⊢ 1 < t₀

Tactic:
simpa using

Goal state after:
X✝ : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X✝
inst✝¹ : TopologicalSpace Y
x y z : X✝
ι : Type u_3
γ✝ : Path x y
X : Type u_4
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ : ℝ
h₁ : ¬t₀ ≤ ↑1
h₃ : t₀ ≤ t₁
⊢ 1 < t₀
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ x : ℝ
⊢ ∀ (h : x ∈ I), (γ.truncate t₀ t₁) ⟨x, h⟩ ∈ range ⇑γ.extend

Tactic:
_hx

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ x : ℝ
_hx : x ∈ I
⊢ (γ.truncate t₀ t₁) ⟨x, _hx⟩ ∈ range ⇑γ.extend
Module: Mathlib.Algebra.Order.Ring.Interval

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
r : R
hr : 0 < r
k m : ℤ
h : r * ↑k ∈ Set.Ioo (r * ↑(m - 1)) (r * ↑(m + 1))
⊢ k = m

Tactic:
simpa using

Goal state after:
R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
r : R
hr : 0 < r
k m : ℤ
h : r * ↑k ∈ Set.Ioo (r * ↑(m - 1)) (r * ↑(m + 1))
⊢ k = m
Module: Mathlib.Analysis.Normed.Group.AddCircle

Goal state before:
pp: p : ℝ
hp : Fact (0 < p)
u : AddCircle p
n : ℕ
hn : ‖u‖ = p * (↑n / ↑(addOrderOf u))
hu : ↑(addOrderOf u) ≠ 0
hu' : n = 0
⊢ u = 0

Tactic:
simpa using

Goal state after:
p : ℝ
hp : Fact (0 < p)
u : AddCircle p
n : ℕ
hn : ‖u‖ = p * (↑n / ↑(addOrderOf u))
hu : ↑(addOrderOf u) ≠ 0
hu' : n = 0
⊢ u = 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse

Goal state before:
pp: x : ℝ
hx : x ∈ Icc (-1) 1
⊢ sin (arcsin x) = x

Tactic:
simpa using

Goal state after:
x : ℝ
hx : x ∈ Icc (-1) 1
⊢ sin ↑(sinOrderIso.symm ⟨x, hx⟩) = x