mathlib-initiative/mathlib-tactics huggingface dataset:
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∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x r
h : ↑r' < r
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : C > 0
hp : ∀ (n : ℕ), ‖p n‖ * ↑r' ^ n ≤ C * a ^ n
y : E
hy : y ∈ Metric.ball 0 ↑r'
n : ℕ
ys : x + y ∈ insert x s
yr' : ‖y‖ < ↑r'
hr'0 : 0 < ↑r'
this✝ : y ∈ Metric.eball 0 r
ya : a * (‖y‖ / ↑r') ≤ a
this : ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n / (1 - a * (‖y‖ / ↑r'))
⊢ ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n / (1 - a)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : 0 < n
v : Fin n → E
⊢ p.applyComposition (Composition.single n hn) v = fun _j ↦ (p n) v

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
c : Composition n
⊢ p.removeZero.applyComposition c = p.applyComposition c

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∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x r
h : ↑r' < r
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : C > 0
hp : ∀ (n : ℕ), ‖p n‖ * ↑r' ^ n ≤ C * a ^ n
y : E
hy : y ∈ Metric.ball 0 ↑r'
n✝ : ℕ
ys : x + y ∈ insert x s
yr' : ‖y‖ < ↑r'
hr'0 : 0 < ↑r'
this : y ∈ Metric.eball 0 r
ya : a * (‖y‖ / ↑r') ≤ a
n : ℕ
⊢ ‖p n‖ * ∏ _i, ‖y‖ = ‖p n‖ * ↑r' ^ n * (‖y‖ / ↑r') ^ n

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
c : Composition n
j : Fin n
v : Fin n → E
z : E
k : Fin c.length
h : k = c.index j
r : Fin (c.blocksFun (c.index j)) → Fin n := ⇑(c.embedding (c.index j))
j' : Fin (c.blocksFun (c.index j)) := c.invEmbedding j
B : Function.update v j z ∘ r = Function.update (v ∘ r) j' z
⊢ (p (c.blocksFun (c.index j))) (Function.update v j z ∘ r) =
    (p (c.blocksFun (c.index j))) (Function.update (v ∘ ⇑(c.embedding (c.index j))) (c.invEmbedding j) z)

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∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x r
h : ↑r' < r
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : C > 0
hp : ∀ (n : ℕ), ‖p n‖ * ↑r' ^ n ≤ C * a ^ n
y : E
hy : y ∈ Metric.ball 0 ↑r'
n✝ : ℕ
ys : x + y ∈ insert x s
yr' : ‖y‖ < ↑r'
hr'0 : 0 < ↑r'
this : y ∈ Metric.eball 0 r
ya : a * (‖y‖ / ↑r') ≤ a
n : ℕ
⊢ ‖p n‖ * ↑r' ^ n * (‖y‖ / ↑r') ^ n ≤ C * a ^ n * (‖y‖ / ↑r') ^ n

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
c : Composition n
j : Fin n
v : Fin n → E
z : E
k : Fin c.length
h : k = c.index j
r : Fin (c.blocksFun (c.index j)) → Fin n := ⇑(c.embedding (c.index j))
j' : Fin (c.blocksFun (c.index j)) := c.invEmbedding j
C : Function.update v (r j') z ∘ r = Function.update (v ∘ r) j' z
⊢ Function.update v j z ∘ r = Function.update (v ∘ r) j' z

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∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x r
h : ↑r' < r
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : C > 0
hp : ∀ (n : ℕ), ‖p n‖ * ↑r' ^ n ≤ C * a ^ n
y : E
hy : y ∈ Metric.ball 0 ↑r'
n✝ : ℕ
ys : x + y ∈ insert x s
yr' : ‖y‖ < ↑r'
hr'0 : 0 < ↑r'
this : y ∈ Metric.eball 0 r
ya : a * (‖y‖ / ↑r') ≤ a
n : ℕ
⊢ C * a ^ n * (‖y‖ / ↑r') ^ n ≤ C * (a * (‖y‖ / ↑r')) ^ n

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∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x r
h : ↑r' < r
⊢ ∃ a ∈ Ioo 0 1,
    ∃ C > 0,
      ∀ y ∈ Metric.ball 0 ↑r',
        ∀ (n : ℕ), x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
c : Composition n
j : Fin n
v : Fin n → E
z : E
k : Fin c.length
h : ¬k = c.index j
r : Fin (c.blocksFun k) → Fin n := ⇑(c.embedding k)
B : Function.update v j z ∘ r = v ∘ r
⊢ (p (c.blocksFun k)) (Function.update v j z ∘ r) = (p (c.blocksFun k)) (v ∘ 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∞
r' : ℝ≥0
hf : HasFPowerSeriesOnBall f p x r
h : ↑r' < r
⊢ ∃ a ∈ Ioo 0 1, ∃ C > 0, ∀ y ∈ Metric.ball 0 ↑r', ∀ (n : ℕ), ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
c : Composition n
j : Fin n
v : Fin n → E
z : E
⊢ p.applyComposition c (Function.update v j z) =
    Function.update (p.applyComposition c v) (c.index j)
      ((p (c.blocksFun (c.index j))) (Function.update (v ∘ ⇑(c.embedding (c.index j))) (c.invEmbedding j) z))

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
n : ℕ
p : FormalMultilinearSeries 𝕜 F G
f : E →L[𝕜] F
c : Composition n
v : Fin n → E
⊢ (p.compContinuousLinearMap f).applyComposition c v = p.applyComposition c (⇑f ∘ v)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousConstSMul 𝕜 E
H : Type u_6
inst✝⁴ : CommRing H
inst✝³ : Algebra 𝕜 H
inst✝² : TopologicalSpace H
inst✝¹ : IsTopologicalRing H
inst✝ : ContinuousConstSMul 𝕜 H
p : FormalMultilinearSeries 𝕜 E H
n : ℕ
c : Composition n
v : Fin n → E
⊢ ∏ i, p.applyComposition c v i = ∏ i, (p (c.blocksFun i)) (v ∘ ⇑(c.embedding i))

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∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x r
h : ↑r' < r
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : C > 0
hp : ∀ y ∈ Metric.ball 0 ↑r', ∀ (n : ℕ), x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n
y : E
hy : y ∈ Metric.ball 0 ↑r'
n : ℕ
ys : x + y ∈ insert x s
⊢ ‖y‖ < ↑r'

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝¹³ : CommRing 𝕜
inst✝¹² : AddCommGroup E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : AddCommGroup G
inst✝⁹ : Module 𝕜 E
inst✝⁸ : Module 𝕜 F
inst✝⁷ : Module 𝕜 G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
n : ℕ
p : FormalMultilinearSeries 𝕜 E F
c : Composition n
f : F [×c.length]→L[𝕜] G
v : Fin n → E
i : Fin n
x y : E
⊢ f (p.applyComposition c (Function.update v i (x + y))) =
    f (p.applyComposition c (Function.update v i x)) + f (p.applyComposition c (Function.update v i y))

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝¹³ : CommRing 𝕜
inst✝¹² : AddCommGroup E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : AddCommGroup G
inst✝⁹ : Module 𝕜 E
inst✝⁸ : Module 𝕜 F
inst✝⁷ : Module 𝕜 G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
n : ℕ
p : FormalMultilinearSeries 𝕜 E F
c✝ : Composition n
f : F [×c✝.length]→L[𝕜] G
v : Fin n → E
i : Fin n
c : 𝕜
x : E
⊢ f (p.applyComposition c✝ (Function.update v i (c • x))) = c • f (p.applyComposition c✝ (Function.update v i x))

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
v : Fin 0 → E
v' : Fin 0 → F
c : Composition 0 := Composition.ones 0
⊢ {c} = Finset.univ

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∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x r
h : ↑r' < r
⊢ ∃ a ∈ Ioo 0 1,
    ∃ C > 0, ∀ y ∈ Metric.ball 0 ↑r', ∀ (n : ℕ), x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
v : Fin 0 → E
v' : Fin 0 → F
⊢ (q.comp p 0) v = (q 0) v'

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
q : FormalMultilinearSeries 𝕜 E F
p : FormalMultilinearSeries 𝕜 E E
⊢ q.comp p 0 = q 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∞
r' : ℝ≥0
hf : HasFPowerSeriesOnBall f p x r
h : ↑r' < r
⊢ ∃ a ∈ Ioo 0 1, ∃ C > 0, ∀ y ∈ Metric.ball 0 ↑r', ∀ (n : ℕ), ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
v : Fin 1 → E
⊢ {Composition.ones 1}.card = Fintype.card (Composition 1)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
v : Fin 1 → E
this : {Composition.ones 1} = Finset.univ
⊢ (Composition.ones 1).length = 1

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
v : Fin 1 → E
this : {Composition.ones 1} = Finset.univ
i : ℕ
hi1 : i < (Composition.ones 1).length
hi2 : i < 1
j : ℕ
_hj1 hj2 : j < 1
⊢ v (Fin.castLE ⋯ ⟨i, hi1⟩) = v ⟨j, hj2⟩

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
v : Fin 1 → E
⊢ (q.comp p 1) v = (q 1) fun _i ↦ (p 1) v

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
n : ℕ
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
r' : ℝ≥0
r'0 : 0 < ↑r'
h : ↑r' < r
a C : ℝ
hp : ∀ y ∈ Metric.ball 0 ↑r', ∀ (n : ℕ), x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n
⊢ ∀ᶠ (x_1 : E) in 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' insert x s] 0,
    ‖f (x + x_1) - p.partialSum n x_1‖ ≤ C * (a / ↑r') ^ n * ‖‖x_1‖ ^ n‖

Tactic:
replace r'0 : 0 < (r' : ℝ) := mod_cast r'0

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
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
n : ℕ
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
r' : ℝ≥0
h : ↑r' < r
a C : ℝ
hp : ∀ y ∈ Metric.ball 0 ↑r', ∀ (n : ℕ), x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n
r'0 : 0 < ↑r'
⊢ ∀ᶠ (x_1 : E) in 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' insert x s] 0,
    ‖f (x + x_1) - p.partialSum n x_1‖ ≤ C * (a / ↑r') ^ n * ‖‖x_1‖ ^ n‖
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : 0 < n
⊢ q.removeZero.comp p n = q.comp p n

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
⊢ q.comp p.removeZero = q.comp p

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
n : ℕ
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
r' : ℝ≥0
h : ↑r' < r
a C : ℝ
hp : ∀ y ∈ Metric.ball 0 ↑r', ∀ (n : ℕ), x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n
r'0 : 0 < ↑r'
y : E
hy : y ∈ (fun x_1 ↦ x + x_1) ⁻¹' insert x s ∩ Metric.ball 0 ↑r'
⊢ x + y ∈ 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
hf : HasFPowerSeriesWithinAt f p s x
n : ℕ
⊢ (fun y ↦ f (x + y) - p.partialSum n y) =O[𝓝[(fun x_1 ↦ x + x_1) ⁻¹' insert x s] 0] fun y ↦ ‖y‖ ^ n

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
p : FormalMultilinearSeries 𝕜 E F
c : Composition n
f : F [×c.length]→L[𝕜] G
v : Fin n → E
⊢ ‖f‖ * ∏ i, ‖p.applyComposition c v i‖ ≤ ‖f‖ * ∏ i, ‖p (c.blocksFun i)‖ * ∏ j, ‖(v ∘ ⇑(c.embedding i)) j‖

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
n : ℕ
⊢ (fun y ↦ f (x + y) - p.partialSum n y) =O[𝓝 0] fun y ↦ ‖y‖ ^ n

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
p : FormalMultilinearSeries 𝕜 E F
c : Composition n
f : F [×c.length]→L[𝕜] G
v : Fin n → E
⊢ ‖f‖ * ∏ i, ‖p (c.blocksFun i)‖ * ∏ j, ‖(v ∘ ⇑(c.embedding i)) j‖ =
    (‖f‖ * ∏ i, ‖p (c.blocksFun i)‖) * ∏ i, ∏ j, ‖(v ∘ ⇑(c.embedding i)) j‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
p : FormalMultilinearSeries 𝕜 E F
c : Composition n
f : F [×c.length]→L[𝕜] G
v : Fin n → E
⊢ (‖f‖ * ∏ i, ‖p (c.blocksFun i)‖) * ∏ i, ∏ j, ‖(v ∘ ⇑(c.embedding i)) j‖ =
    (‖f‖ * ∏ i, ‖p (c.blocksFun i)‖) * ∏ i, ‖v i‖

Tactic:
by

Goal state after: