mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Instances.Sign

Goal state before:
pp: α : Type u_1
inst✝³ : Zero α
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
h : a ≠ 0
⊢ ContinuousAt (⇑SignType.sign) a

Tactic:
by

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

Goal state before:
pp: p x : ℝ
this : ∀ (x : ℝ), ‖↑x‖ = |x - ↑(round x)|
⊢ ‖↑x‖ = |x - ↑(round (p⁻¹ * x)) * p|

Tactic:
by

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

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

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
ι : Type u_4
inst✝ : TopologicalSpace ι
a b c : ι → X
γ₁ : (t : ι) → Path (a t) (b t)
h₁ : Continuous[instTopologicalSpaceProd, inst✝¹] ↿γ₁
γ₂ : (t : ι) → Path (b t) (c t)
h₂ : Continuous[instTopologicalSpaceProd, inst✝¹] ↿γ₂
h₁' : Continuous[instTopologicalSpaceProd, inst✝¹] ↿fun t ↦ ⇑(γ₁ t).extend
h₂' : Continuous[instTopologicalSpaceProd, inst✝¹] ↿fun t ↦ ⇑(γ₂ t).extend
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] (Prod.map id fun x ↦ 2 * ↑x - 1)

Tactic:
by

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

Goal state before:
pp: p x : ℝ
⊢ ‖↑x‖ = |x - ↑(round (p⁻¹ * x)) * p|

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
b : ℝ
f : ℕ → ℝ
z : ℕ → E
hfa : Monotone f
hf0 : Tendsto f atTop (𝓝 0)
hgb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b
⊢ CauchySeq fun n ↦ ∑ i ∈ Finset.range n, f i • z i

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
b : ℝ
f : ℕ → ℝ
z : ℕ → E
hfa : Antitone f
hf0 : Tendsto f atTop (𝓝 0)
hzb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b
hfa' : Monotone fun n ↦ -f n
⊢ Tendsto (fun n ↦ -f n) atTop (𝓝 0)

Tactic:
by

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

Goal state before:
pp: p : ℝ
hp : 0 < p
x : ℝ
⊢ ‖↑x‖ = p * |p⁻¹ * x - ↑(round (p⁻¹ * x))|

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
b : ℝ
f : ℕ → ℝ
z : ℕ → E
hfa : Antitone f
hf0 : Tendsto f atTop (𝓝 0)
hzb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b
⊢ CauchySeq fun n ↦ ∑ i ∈ Finset.range n, f i • z i

Tactic:
by

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

Goal state before:
pp: p : ℝ
x : AddCircle p
hp : p ≠ 0
⊢ ‖x‖ ≤ |p| / 2

Tactic:
by

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

Goal state before:
pp: n : ℕ
⊢ ‖∑ i ∈ Finset.range n, (-1) ^ i‖ ≤ 1

Tactic:
by

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

Goal state before:
pp: f : ℕ → ℝ
hfa : Monotone f
hf0 : Tendsto f atTop (𝓝 0)
⊢ CauchySeq fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i

Tactic:
by

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

Goal state before:
pp: f : ℕ → ℝ
hfa : Antitone f
hf0 : Tendsto f atTop (𝓝 0)
⊢ CauchySeq fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
ι : Type u_4
inst✝ : TopologicalSpace ι
a b c : ι → X
γ₁ : (t : ι) → Path (a t) (b t)
h₁ : Continuous[instTopologicalSpaceProd, inst✝¹] ↿γ₁
γ₂ : (t : ι) → Path (b t) (c t)
h₂ : Continuous[instTopologicalSpaceProd, inst✝¹] ↿γ₂
⊢ Continuous[instTopologicalSpaceProd, inst✝¹] ↿fun t ↦ (γ₁ t).trans (γ₂ t)

Tactic:
by

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

Goal state before:
pp: p : ℝ
hp : p ≠ 0
⊢ p / 2 - p = -(p / 2)

Tactic:
by

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

Goal state before:
pp: p : ℝ
⊢ ‖↑(p / 2)‖ = |p| / 2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
f : Y → Path x y
g : Y → Path y z
⊢ Continuous[inst✝, _] f → Continuous[inst✝, _] g → Continuous[inst✝, _] fun t ↦ (f t).trans (g t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
⊢ Continuous fun ρ ↦ ρ.1.trans ρ.2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ : Path x y
a₁ a₂ a₃ : X
b₁ b₂ b₃ : Y
γ₁ : Path a₁ a₂
γ₂ : Path b₁ b₂
⊢ (γ₁.prodMk γ₂.toContinuousMap).toFun 0 = (a₁, b₁)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ : Path x y
a₁ a₂ a₃ : X
b₁ b₂ b₃ : Y
γ₁ : Path a₁ a₂
γ₂ : Path b₁ b₂
⊢ (γ₁.prodMk γ₂.toContinuousMap).toFun 1 = (a₂, b₂)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
a₁ a₂ a₃ : X
b₁ b₂ b₃ : Y
γ₁ : Path a₁ a₂
δ₁ : Path a₂ a₃
γ₂ : Path b₁ b₂
δ₂ : Path b₂ b₃
⊢ (γ₁.prod γ₂).trans (δ₁.prod δ₂) = (γ₁.trans δ₁).prod (γ₂.trans δ₂)

Tactic:
by

Goal state after:
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
χ : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (χ i)
as bs cs : (i : ι) → χ i
γ : (i : ι) → Path (as i) (bs i)
⊢ (ContinuousMap.pi fun i ↦ (γ i).toContinuousMap).toFun 0 = as

Tactic:
by

Goal state after:
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
χ : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (χ i)
as bs cs : (i : ι) → χ i
γ : (i : ι) → Path (as i) (bs i)
⊢ (ContinuousMap.pi fun i ↦ (γ i).toContinuousMap).toFun 1 = bs

Tactic:
by

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

Goal state before:
pp: p x : ℝ
hp : p ≠ 0
hx : |x| ≤ |p| / 2
this : ∀ (p : ℝ), 0 < p → |x| ≤ p / 2 → ‖↑x‖ = |x|
⊢ ‖↑x‖ = |x|

Tactic:
by

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

Goal state before:
pp: p✝ x : ℝ
hp✝ : p✝ ≠ 0
p : ℝ
hp : 0 < p
hx : |x| ≤ p / 2
hx' : x ≠ p / 2
this : round (p⁻¹ * x) = 0
⊢ ‖↑x‖ = |x|

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: ι : Type u_3
χ : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (χ i)
as bs cs : (i : ι) → χ i
γ₀ : (i : ι) → Path (as i) (bs i)
γ₁ : (i : ι) → Path (bs i) (cs i)
⊢ (Path.pi γ₀).trans (Path.pi γ₁) = Path.pi fun i ↦ (γ₀ i).trans (γ₁ i)

Tactic:
by

Goal state after:
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
⊢ Continuous fun s ↦ min (max (↑s) t₀) t₁

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k n : ℕ
⊢ 2 * n ≤ 2 * n + 1

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k : ℕ
⊢ Antitone fun n ↦ ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k : ℕ
ha : Antitone fun n ↦ ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i
n : ℕ
⊢ id n ≤ 2 * n

Tactic:
by

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

Goal state before:
pp: p x : ℝ
hp : p ≠ 0
⊢ ‖↑x‖ = |x| ↔ |x| ≤ |p| / 2

Tactic:
by

Goal state after:
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:
by

Goal state after:
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:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k : ℕ
⊢ l ≤ ∑ i ∈ Finset.range (2 * k), (-1) ^ i * f i

Tactic:
by

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

Goal state before:
pp: x ε : ℝ
⊢ QuotientAddGroup.mk ⁻¹' closedBall (↑x) ε = closedBall x ε

Tactic:
by

Goal state after:
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
⊢ γ.extend (min (max (↑0) t₀) t₁) = γ.extend (min t₀ t₁)

Tactic:
by

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

Goal state before:
pp: p x ε : ℝ
⊢ QuotientAddGroup.mk ⁻¹' closedBall (↑x) ε = ⋃ z, closedBall (x + z • p) ε

Tactic:
by

Goal state after:
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:
by

Goal state after:
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
⊢ γ.extend (min (max (↑1) t₀) t₁) = γ.extend t₁

Tactic:
by

Goal state after:
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
h : t₀ ≤ t₁
⊢ γ.extend t₀ = γ.extend (min t₀ t₁)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ : ℝ
⊢ range ⇑(γ.truncate t₀ t₁) ⊆ range ⇑γ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t : ℝ
⊢ Continuous fun x ↦ (t, x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t : ℝ
⊢ Continuous[_, inst✝] ↿(γ.truncate t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ✝ : Path x y
a b : X
γ : Path a b
t : ℝ
⊢ γ.extend (min t t) = γ.extend t

Tactic:
by

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

Goal state before:
pp: x ε : ℝ
s : Set ℝ
hs : s ⊆ closedBall x (|0| / 2)
hε : 0 ≤ ε
⊢ |0| / 2 ≤ ε

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t : ℝ
⊢ γ.truncate t t = (refl (γ.extend t)).cast ⋯ ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ✝ : Path x y
a b : X
γ : Path a b
⊢ γ.extend (min 0 0) = a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
⊢ γ.truncate 0 0 = (refl a).cast ⋯ ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ✝ : Path x y
a b : X
γ : Path a b
⊢ γ.extend (min 1 1) = b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
⊢ γ.truncate 1 1 = (refl b).cast ⋯ ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ✝ : Path x y
a b : X
γ : Path a b
⊢ γ.extend (min 0 1) = a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ✝ : Path x y
a b : X
γ : Path a b
⊢ γ.extend 1 = b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
⊢ γ.truncate 0 1 = γ.cast ⋯ ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ✝ γ : Path x y
f : ↑I → ↑I
hfcont : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ Continuous[_, inst✝¹] (⇑γ ∘ f)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ✝ γ : Path x y
f : ↑I → ↑I
hfcont : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ (⇑γ ∘ f) 0 = x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ✝ γ : Path x y
f : ↑I → ↑I
hfcont : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ (⇑γ ∘ f) 1 = y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
x y : X
γ : Path x y
⊢ γ.reparam id ⋯ ⋯ ⋯ = γ

Tactic:
by

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

Goal state before:
pp: p x ε : ℝ
s : Set ℝ
hs : s ⊆ closedBall x (|p| / 2)
hε : ε < |p| / 2
this : ∀ (z : ℤ), closedBall (x + z • p) ε ∩ s = if z = 0 then closedBall x ε ∩ s else ∅
⊢ QuotientAddGroup.mk ⁻¹' closedBall (↑x) ε ∩ s = if ε < |p| / 2 then closedBall x ε ∩ s else s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
x y : X
γ : Path x y
f : ↑I → ↑I
hfcont : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
t : ↑I
⊢ Continuous (IccExtend ⋯ f)

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k n : ℕ
⊢ 2 * n + 1 ≤ 2 * n + 1 + 1

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k : ℕ
⊢ Monotone fun n ↦ ∑ i ∈ Finset.range (2 * n + 1), (-1) ^ i * f i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
x y : X
γ : Path x y
f : ↑I → ↑I
hfcont : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ range f = univ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
x y : X
γ : Path x y
f : ↑I → ↑I
hfcont : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ range ⇑(γ.reparam f hfcont hf₀ hf₁) = range ⇑γ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
x : X
f : ↑I → ↑I
hfcont : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ (refl x).reparam f hfcont hf₀ hf₁ = refl x

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k : ℕ
hm : Monotone fun n ↦ ∑ i ∈ Finset.range (2 * n + 1), (-1) ^ i * f i
n : ℕ
⊢ id n ≤ 2 * n + 1

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k : ℕ
⊢ ∑ i ∈ Finset.range (2 * k + 1), (-1) ^ i * f i ≤ l

Tactic:
by

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

Goal state before:
pp: p x ε : ℝ
s : Set ℝ
hε : ε < |p| / 2
z : ℤ
hs : s ⊆ Icc (x - |p| / 2) (x + |p| / 2)
hz : z ≠ 0
y : ℝ
hy₀ : y ∈ s
hy₁ : x + ↑z * p - ε ≤ y
hy₂ : y ≤ x + ↑z * p + ε
hy₃ : x - |p| / 2 ≤ y
hy₄ : y ≤ x + |p| / 2
hp : 0 < p
hz' : ↑z ≤ -1
⊢ ↑z * p ≤ -p

Tactic:
by

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

Goal state before:
pp: p x ε : ℝ
s : Set ℝ
hε : ε < |p| / 2
z : ℤ
hs : s ⊆ Icc (x - |p| / 2) (x + |p| / 2)
hz : z ≠ 0
y : ℝ
hy₀ : y ∈ s
hy₁ : x + ↑z * p - ε ≤ y
hy₂ : y ≤ x + ↑z * p + ε
hy₃ : x - |p| / 2 ≤ y
hy₄ : y ≤ x + |p| / 2
hp : 0 < p
hz' : 1 ≤ ↑z
⊢ p ≤ ↑z * p

Tactic:
by

Goal state after:
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 n : ℤ
h : r * ↑k ∈ Set.Ioo (r * ↑(m - 1)) (r * ↑(n + 1))
⊢ k ∈ Finset.Icc m n

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k n : ℕ
⊢ 2 * n ≤ 2 * n + 1

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k : ℕ
⊢ Monotone fun n ↦ ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i

Tactic:
by

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

Goal state before:
pp: p x ε : ℝ
s : Set ℝ
hε : ε < |p| / 2
z : ℤ
hs : s ⊆ Icc (x - |p| / 2) (x + |p| / 2)
hz : z ≠ 0
y : ℝ
hy₀ : y ∈ s
hy₁ : x + ↑z * p - ε ≤ y
hy₂ : y ≤ x + ↑z * p + ε
hy₃ : x - |p| / 2 ≤ y
hy₄ : y ≤ x + |p| / 2
hp : p < 0
hz' : ↑z ≤ -1
⊢ -p ≤ ↑z * p

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k : ℕ
hm : Monotone fun n ↦ ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i
n : ℕ
⊢ id n ≤ 2 * n

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k : ℕ
⊢ ∑ i ∈ Finset.range (2 * k), (-1) ^ i * f i ≤ l

Tactic:
by

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

Goal state before:
pp: p x ε : ℝ
s : Set ℝ
hε : ε < |p| / 2
z : ℤ
hs : s ⊆ Icc (x - |p| / 2) (x + |p| / 2)
hz : z ≠ 0
y : ℝ
hy₀ : y ∈ s
hy₁ : x + ↑z * p - ε ≤ y
hy₂ : y ≤ x + ↑z * p + ε
hy₃ : x - |p| / 2 ≤ y
hy₄ : y ≤ x + |p| / 2
hp : p < 0
hz' : 1 ≤ ↑z
⊢ ↑z * p ≤ p

Tactic:
by

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

Goal state before:
pp: p x ε : ℝ
s : Set ℝ
hs : s ⊆ closedBall x (|p| / 2)
⊢ QuotientAddGroup.mk ⁻¹' closedBall (↑x) ε ∩ s = if ε < |p| / 2 then closedBall x ε ∩ s else s

Tactic:
by

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

Goal state before:
pp: p : ℝ
hp : Fact (0 < p)
m n : ℕ
⊢ p⁻¹ * (↑m / ↑n * p) = ↑m / ↑n

Tactic:
by

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

Goal state before:
pp: p : ℝ
hp : Fact (0 < p)
m n : ℕ
⊢ ‖↑(↑m / ↑n * p)‖ = p * (↑(min (m % n) (n - m % n)) / ↑n)

Tactic:
by

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

Goal state before:
pp: p : ℝ
hp : Fact (0 < p)
u : AddCircle p
hu : IsOfFinAddOrder u
⊢ ∃ k, ‖u‖ = p * (↑k / ↑(addOrderOf u))

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k n : ℕ
⊢ 2 * n + 1 ≤ 2 * n + 2

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k : ℕ
⊢ Antitone fun n ↦ ∑ i ∈ Finset.range (2 * n + 1), (-1) ^ i * f i

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k : ℕ
ha : Antitone fun n ↦ ∑ i ∈ Finset.range (2 * n + 1), (-1) ^ i * f i
n : ℕ
⊢ id n ≤ 2 * n + 1

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : Ring E
inst✝³ : PartialOrder E
inst✝² : IsOrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k : ℕ
⊢ l ≤ ∑ i ∈ Finset.range (2 * k + 1), (-1) ^ i * f i

Tactic:
by

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

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

Tactic:
by

Goal state after: