mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c c' : ℕ → 𝕜
⊢ ofScalars E (c + c') = ofScalars E c + ofScalars E c'

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c c' : ℕ → 𝕜
⊢ ofScalars E (c - c') = ofScalars E c - ofScalars E c'

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c : ℕ → 𝕜
x : 𝕜
⊢ ofScalars E (x • c) = x • ofScalars E c

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
⊢ ∞ ≤ p.radius

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
y : E
x✝ : y ∈ Metric.eball 0 ∞
⊢ HasSum (fun n ↦ (p n) fun x ↦ y) (p.sum (0 + y))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c : ℕ → 𝕜
⊢ (ofScalars E c).compContinuousLinearMap (-ContinuousLinearMap.id 𝕜 E) = ofScalars E fun k ↦ (-1) ^ k * c k

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    p.sum

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c : ℕ → 𝕜
f : E →L[𝕜] E
⊢ (ofScalars E c).compContinuousLinearMap (-f) = (ofScalars E fun k ↦ (-1) ^ k * c k).compContinuousLinearMap f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c : ℕ → 𝕜
x : E
n : ℕ
⊢ ((ofScalars E c n) fun x_1 ↦ x) = c n • x ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c : ℕ → 𝕜
x : E
⊢ (fun n ↦ (ofScalars E c n) fun x_1 ↦ x) = fun n ↦ c n • x ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c : ℕ → 𝕜
⊢ ofScalarsSum c 0 = c 0 • 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Field 𝕜
inst✝⁴ : Ring E
inst✝³ : Algebra 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalRing E
c : ℕ → 𝕜
inst✝ : Subsingleton E
x : E
⊢ ofScalarsSum c x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Field 𝕜
inst✝⁴ : Ring E
inst✝³ : Algebra 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalRing E
c : ℕ → 𝕜
inst✝ : T2Space E
x : E
⊢ ofScalarsSum c (MulOpposite.op x) = MulOpposite.op (ofScalarsSum c x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k l : ℕ
s : Finset (Fin (k + l))
hs : s.card = l
hkl : n ≤ k + l
⊢ p.changeOriginSeriesTerm k l s hs = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Field 𝕜
inst✝⁴ : Ring E
inst✝³ : Algebra 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalRing E
c : ℕ → 𝕜
inst✝ : T2Space E
x : Eᵐᵒᵖ
⊢ ofScalarsSum c (MulOpposite.unop x) = MulOpposite.unop (ofScalarsSum c x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
n : ℕ
⊢ ‖ofScalars E c n‖ = ‖c n‖ * ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n E‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
n : ℕ
hn : n > 0
⊢ ‖ofScalars E c n‖ ≤ ‖c n‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
n : ℕ
inst✝ : NormOneClass E
⊢ ‖ofScalars E c n‖ = ‖c n‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k : ℕ
⊢ ∀ {m : ℕ}, n ≤ k + m → p.changeOriginSeries k m = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesOnBall f p x r
y z : E
hz : z ∈ Metric.eball 0 r
⊢ HasSum (fun n ↦ (p n) fun x ↦ z) (f (x + y + z - y))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
c : ℕ → 𝕜
r r' : ℝ≥0
hr' : r' ≠ 0
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)
⊢ Tendsto (fun n ↦ ‖‖c (n + 1)‖ * ↑r' ^ (n + 1)‖ / ‖‖c n‖ * ↑r' ^ n‖) atTop (𝓝 ↑(r' * r))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
y z : E
hz1 : x + y + z ∈ insert (x + y) (s + {y})
hz2 : z ∈ Metric.eball 0 r
⊢ x + z ∈ insert x s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
y z : E
hz1 : x + y + z ∈ insert (x + y) (s + {y})
hz2 : z ∈ Metric.eball 0 r
⊢ HasSum (fun n ↦ (p n) fun x ↦ z) (f (x + y + z - y))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
hf : HasFPowerSeriesAt f p x
y : E
⊢ HasFPowerSeriesAt (fun z ↦ f (z - y)) p (x + y)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
hf : HasFPowerSeriesWithinAt f p s x
y : E
⊢ HasFPowerSeriesWithinAt (fun z ↦ f (z - y)) p (s + {y}) (x + y)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k : ℕ
x : E
x✝ : Fin k → E
m : ℕ
hm : n - k ≤ m
⊢ n ≤ k + m

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
hf : AnalyticAt 𝕜 f x
y : E
⊢ AnalyticAt 𝕜 (fun z ↦ f (z - y)) (x + y)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k : ℕ
⊢ (p.changeOriginSeries k).sum = (p.changeOriginSeries k).partialSum (n - k)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: case neg.refine_1.refine_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)
r' : ℝ≥0
hr' : r' * r < 1
hrz : ¬r' = 0
⊢ Tendsto (fun n ↦ ‖‖c (n + 1)‖ * ↑r' ^ (n + 1)‖ / ‖‖c n‖ * ↑r' ^ n‖) atTop (𝓝 ↑(r' * r))

Tactic:
exact tendsto_succ_norm_div_norm c hrz hc

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
hf : AnalyticOnNhd 𝕜 f s
y x : E
hx : x + -y ∈ s
⊢ x - y ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k : ℕ
hk : n ≤ k
⊢ p.changeOrigin x k = 0

Tactic:
rw [changeOrigin, p.changeOriginSeries_sum_eq_partialSum_of_finite hn]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k : ℕ
hk : n ≤ k
⊢ (p.changeOriginSeries k).partialSum (n - k) x = 0
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
hf : AnalyticOnNhd 𝕜 f s
y : E
⊢ AnalyticOnNhd 𝕜 (fun z ↦ f (z - y)) (s + {y})

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
x : E
hf : AnalyticWithinAt 𝕜 f s x
y : E
⊢ AnalyticWithinAt 𝕜 (fun z ↦ f (z - y)) (s + {y}) (x + y)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
hf : AnalyticOn 𝕜 f s
y x : E
hx : x + -y ∈ s
⊢ x - y ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
hf : AnalyticOn 𝕜 f s
y : E
⊢ AnalyticOn 𝕜 (fun z ↦ f (z - y)) (s + {y})

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
y : E
hy : y ∈ insert x s ∩ Metric.eball x r
⊢ y - x ∈ Metric.eball 0 r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)
⊢ ↑r⁻¹ ≤ (ofScalars E c).radius

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k : ℕ
hk : n ≤ k
⊢ p.changeOrigin x k = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
y : E
hy : y ∈ insert x s ∩ Metric.eball x r
this : y - x ∈ Metric.eball 0 r
⊢ x + (y - x) ∈ insert x s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
y : E
hy : y ∈ insert x s ∩ Metric.eball x r
⊢ HasSum (fun n ↦ (p n) fun x_1 ↦ y - x) (f y)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n k : ℕ
hn : ∀ (m : ℕ), n + k ≤ m → p m = 0
x✝ : ℕ
hm : n ≤ x✝
⊢ n + k ≤ k + x✝

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesOnBall f p x r
y : E
hy : y ∈ Metric.eball x r
⊢ y - x ∈ Metric.eball 0 r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesOnBall f p x r
y : E
hy : y ∈ Metric.eball x r
⊢ HasSum (fun n ↦ (p n) fun x_1 ↦ y - x) (f y)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)
r' : ℝ≥0
hr' : ↑r' < (ofScalars E c).radius
this : 1 < r' * r
⊢ r' ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)
⊢ (ofScalars E c).radius = ↑r⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n ↦ ‖c n‖ / ‖c n.succ‖) atTop (𝓝 ↑r)
this : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 (↑r)⁻¹)
⊢ (ofScalars E c).radius = ↑r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n ↦ ‖c n‖ / ‖c n.succ‖) atTop (𝓝 ↑r)
⊢ (ofScalars E c).radius = ↑r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : EqOn g f (s ∩ Metric.eball x r)
h'' : g x = f x
⊢ HasFPowerSeriesWithinOnBall g p s x r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : EqOn g f (insert x s ∩ Metric.eball x r)
y : E
hy : x + y ∈ insert x s
h'y : y ∈ Metric.eball 0 r
⊢ x + y ∈ Metric.eball x r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : EqOn g f (insert x s ∩ Metric.eball x r)
⊢ HasFPowerSeriesWithinOnBall g p s x r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
x y : E
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=
  fun s ↦ ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 ↦ x) fun x ↦ y
⊢ (Function.support f).Finite

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
h' : g =ᶠ[𝓝[s] x] f
h'' : g x = f x
r : ℝ≥0∞
hr : HasFPowerSeriesWithinOnBall f p s x r
ε : ℝ≥0∞
εpos : ε > 0
hε : Metric.eball x ε ∩ s ⊆ {y | g y = f y}
r' : ℝ≥0∞ := min r ε
⊢ 0 < r'

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
⊢ (ofScalars E c).radius = ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop
this : (ofScalars E c).radius ≤ 0
⊢ (ofScalars E c).radius = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop
r : ℝ≥0
hr : ↑r < (ofScalars E c).radius
this : 0 < r
⊢ 1 < 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
x y : E
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=
  fun s ↦ ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 ↦ x) fun x ↦ y
finsupp : (Function.support f).Finite
k l : ℕ
⊢ HasSum (fun x ↦ f ⟨k, ⟨l, x⟩⟩) (((p.changeOriginSeries k l) fun x_1 ↦ x) fun x ↦ y)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
h : HasFPowerSeriesWithinAt f p s x
h' : g =ᶠ[𝓝[s] x] f
h'' : g x = f x
⊢ HasFPowerSeriesWithinAt g p s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesOnBall f p x r
hg : EqOn f g (Metric.eball x r)
y : E
hy : y ∈ Metric.eball 0 r
⊢ HasSum (fun n ↦ (p n) fun x ↦ y) (g (x + y))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
hf : HasFPowerSeriesAt f p x
hg : f =ᶠ[𝓝 x] g
⊢ HasFPowerSeriesAt g p x

Tactic:
by

Goal state after: