l3lab/ntp-mathlib huggingface dataset:
lemma HasFPowerSeriesAt.prod {e : E} {f : E → F} {g : E → G}
    {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
    (hf : HasFPowerSeriesAt f p e) (hg : HasFPowerSeriesAt g q e) :
    HasFPowerSeriesAt (fun x ↦ (f x, g x)) (p.prod q) e := by
  rcases hf with ⟨_, hf⟩
  rcases hg with ⟨_, hg⟩
  

Proof state:
case intro.intro
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
e : E
f : E → F
g : E → G
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜 E G
w✝¹ : ℝ≥0∞
hf : HasFPowerSeriesOnBall f p e w✝¹
w✝ : ℝ≥0∞
hg : HasFPowerSeriesOnBall g q e w✝
⊢ HasFPowerSeriesAt (fun x => (f x, g x)) (FormalMultilinearSeries.prod p q) e

Next tactic:
exact ⟨_, hf.prod hg⟩
/-- The Cartesian product of analytic functions is analytic. -/
lemma AnalyticAt.prod {e : E} {f : E → F} {g : E → G}
    (hf : AnalyticAt 𝕜 f e) (hg : AnalyticAt 𝕜 g e) :
    AnalyticAt 𝕜 (fun x ↦ (f x, g x)) e := by
  

Proof state:
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
e : E
f : E → F
g : E → G
hf : AnalyticAt 𝕜 f e
hg : AnalyticAt 𝕜 g e
⊢ AnalyticAt 𝕜 (fun x => (f x, g x)) e

Next tactic:
rcases hf with ⟨_, hf⟩
/-- The Cartesian product of analytic functions is analytic. -/
lemma AnalyticAt.prod {e : E} {f : E → F} {g : E → G}
    (hf : AnalyticAt 𝕜 f e) (hg : AnalyticAt 𝕜 g e) :
    AnalyticAt 𝕜 (fun x ↦ (f x, g x)) e := by
  rcases hf with ⟨_, hf⟩
  

Proof state:
case intro
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
e : E
f : E → F
g : E → G
hg : AnalyticAt 𝕜 g e
w✝ : FormalMultilinearSeries 𝕜 E F
hf : HasFPowerSeriesAt f w✝ e
⊢ AnalyticAt 𝕜 (fun x => (f x, g x)) e

Next tactic:
rcases hg with ⟨_, hg⟩
/-- The Cartesian product of analytic functions is analytic. -/
lemma AnalyticAt.prod {e : E} {f : E → F} {g : E → G}
    (hf : AnalyticAt 𝕜 f e) (hg : AnalyticAt 𝕜 g e) :
    AnalyticAt 𝕜 (fun x ↦ (f x, g x)) e := by
  rcases hf with ⟨_, hf⟩
  rcases hg with ⟨_, hg⟩
  

Proof state:
case intro.intro
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
e : E
f : E → F
g : E → G
w✝¹ : FormalMultilinearSeries 𝕜 E F
hf : HasFPowerSeriesAt f w✝¹ e
w✝ : FormalMultilinearSeries 𝕜 E G
hg : HasFPowerSeriesAt g w✝ e
⊢ AnalyticAt 𝕜 (fun x => (f x, g x)) e

Next tactic:
exact ⟨_, hf.prod hg⟩
/-- Powers of analytic functions (into a normed `𝕜`-algebra) are analytic. -/
lemma AnalyticAt.pow {f : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (n : ℕ) :
    AnalyticAt 𝕜 (fun x ↦ f x ^ n) z := by
  

Proof state:
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
f : E → A
z : E
hf : AnalyticAt 𝕜 f z
n : ℕ
⊢ AnalyticAt 𝕜 (fun x => f x ^ n) z

Next tactic:
induction n with
  | zero =>
    simp only [Nat.zero_eq, pow_zero]
    apply analyticAt_const
  | succ m hm =>
    simp only [pow_succ]
    exact hf.mul hm
/-- Powers of analytic functions (into a normed `𝕜`-algebra) are analytic. -/
lemma AnalyticAt.pow {f : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (n : ℕ) :
    AnalyticAt 𝕜 (fun x ↦ f x ^ n) z := by
  induction n with
  

Proof state:
case zero
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
f : E → A
z : E
hf : AnalyticAt 𝕜 f z
⊢ AnalyticAt 𝕜 (fun x => f x ^ Nat.zero) z

Next tactic:
| zero =>
    simp only [Nat.zero_eq, pow_zero]
    apply analyticAt_const
/-- Powers of analytic functions (into a normed `𝕜`-algebra) are analytic. -/
lemma AnalyticAt.pow {f : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (n : ℕ) :
    AnalyticAt 𝕜 (fun x ↦ f x ^ n) z := by
  induction n with
  | zero =>
    

Proof state:
case zero
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
f : E → A
z : E
hf : AnalyticAt 𝕜 f z
⊢ AnalyticAt 𝕜 (fun x => f x ^ Nat.zero) z

Next tactic:
simp only [Nat.zero_eq, pow_zero]
/-- Powers of analytic functions (into a normed `𝕜`-algebra) are analytic. -/
lemma AnalyticAt.pow {f : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (n : ℕ) :
    AnalyticAt 𝕜 (fun x ↦ f x ^ n) z := by
  induction n with
  | zero =>
    simp only [Nat.zero_eq, pow_zero]
    

Proof state:
case zero
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
f : E → A
z : E
hf : AnalyticAt 𝕜 f z
⊢ AnalyticAt 𝕜 (fun x => 1) z

Next tactic:
apply analyticAt_const
/-- Powers of analytic functions (into a normed `𝕜`-algebra) are analytic. -/
lemma AnalyticAt.pow {f : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (n : ℕ) :
    AnalyticAt 𝕜 (fun x ↦ f x ^ n) z := by
  induction n with
  | zero =>
    simp only [Nat.zero_eq, pow_zero]
    apply analyticAt_const
  

Proof state:
case succ
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
f : E → A
z : E
hf : AnalyticAt 𝕜 f z
m : ℕ
hm : AnalyticAt 𝕜 (fun x => f x ^ m) z
⊢ AnalyticAt 𝕜 (fun x => f x ^ Nat.succ m) z

Next tactic:
| succ m hm =>
    simp only [pow_succ]
    exact hf.mul hm
/-- Powers of analytic functions (into a normed `𝕜`-algebra) are analytic. -/
lemma AnalyticAt.pow {f : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (n : ℕ) :
    AnalyticAt 𝕜 (fun x ↦ f x ^ n) z := by
  induction n with
  | zero =>
    simp only [Nat.zero_eq, pow_zero]
    apply analyticAt_const
  | succ m hm =>
    

Proof state:
case succ
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
f : E → A
z : E
hf : AnalyticAt 𝕜 f z
m : ℕ
hm : AnalyticAt 𝕜 (fun x => f x ^ m) z
⊢ AnalyticAt 𝕜 (fun x => f x ^ Nat.succ m) z

Next tactic:
simp only [pow_succ]
/-- Powers of analytic functions (into a normed `𝕜`-algebra) are analytic. -/
lemma AnalyticAt.pow {f : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (n : ℕ) :
    AnalyticAt 𝕜 (fun x ↦ f x ^ n) z := by
  induction n with
  | zero =>
    simp only [Nat.zero_eq, pow_zero]
    apply analyticAt_const
  | succ m hm =>
    simp only [pow_succ]
    

Proof state:
case succ
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
f : E → A
z : E
hf : AnalyticAt 𝕜 f z
m : ℕ
hm : AnalyticAt 𝕜 (fun x => f x ^ m) z
⊢ AnalyticAt 𝕜 (fun x => f x * f x ^ m) z

Next tactic:
exact hf.mul hm
lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
    (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
    (formalMultilinearSeries_geometric 𝕜 A).radius = 1 := by
  

Proof state:
α : Type u_1
𝕜✝ : Type u_2
inst✝¹⁶ : NontriviallyNormedField 𝕜✝
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace 𝕜✝ E
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜✝ F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜✝ G
inst✝⁹ : NormedAddCommGroup H
inst✝⁸ : NormedSpace 𝕜✝ H
𝕝 : Type u_7
inst✝⁷ : NontriviallyNormedField 𝕝
inst✝⁶ : NormedAlgebra 𝕜✝ 𝕝
A✝ : Type u_8
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜✝ A✝
𝕜 : Type u_10
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_9
inst✝² : NormedRing A
inst✝¹ : NormOneClass A
inst✝ : NormedAlgebra 𝕜 A
⊢ FormalMultilinearSeries.radius (formalMultilinearSeries_geometric 𝕜 A) = 1

Next tactic:
apply le_antisymm
lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
    (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
    (formalMultilinearSeries_geometric 𝕜 A).radius = 1 := by
  apply le_antisymm
  · 

Proof state:
case a
α : Type u_1
𝕜✝ : Type u_2
inst✝¹⁶ : NontriviallyNormedField 𝕜✝
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace 𝕜✝ E
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜✝ F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜✝ G
inst✝⁹ : NormedAddCommGroup H
inst✝⁸ : NormedSpace 𝕜✝ H
𝕝 : Type u_7
inst✝⁷ : NontriviallyNormedField 𝕝
inst✝⁶ : NormedAlgebra 𝕜✝ 𝕝
A✝ : Type u_8
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜✝ A✝
𝕜 : Type u_10
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_9
inst✝² : NormedRing A
inst✝¹ : NormOneClass A
inst✝ : NormedAlgebra 𝕜 A
⊢ FormalMultilinearSeries.radius (formalMultilinearSeries_geometric 𝕜 A) ≤ 1

Next tactic:
refine le_of_forall_nnreal_lt (fun r hr ↦ ?_)
lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
    (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
    (formalMultilinearSeries_geometric 𝕜 A).radius = 1 := by
  apply le_antisymm
  · refine le_of_forall_nnreal_lt (fun r hr ↦ ?_)
    

Proof state:
case a
α : Type u_1
𝕜✝ : Type u_2
inst✝¹⁶ : NontriviallyNormedField 𝕜✝
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace 𝕜✝ E
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜✝ F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜✝ G
inst✝⁹ : NormedAddCommGroup H
inst✝⁸ : NormedSpace 𝕜✝ H
𝕝 : Type u_7
inst✝⁷ : NontriviallyNormedField 𝕝
inst✝⁶ : NormedAlgebra 𝕜✝ 𝕝
A✝ : Type u_8
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜✝ A✝
𝕜 : Type u_10
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_9
inst✝² : NormedRing A
inst✝¹ : NormOneClass A
inst✝ : NormedAlgebra 𝕜 A
r : ℝ≥0
hr : ↑r < FormalMultilinearSeries.radius (formalMultilinearSeries_geometric 𝕜 A)
⊢ ↑r ≤ 1

Next tactic:
rw [← coe_one, ENNReal.coe_le_coe]
lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
    (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
    (formalMultilinearSeries_geometric 𝕜 A).radius = 1 := by
  apply le_antisymm
  · refine le_of_forall_nnreal_lt (fun r hr ↦ ?_)
    rw [← coe_one, ENNReal.coe_le_coe]
    

Proof state:
case a
α : Type u_1
𝕜✝ : Type u_2
inst✝¹⁶ : NontriviallyNormedField 𝕜✝
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace 𝕜✝ E
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜✝ F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜✝ G
inst✝⁹ : NormedAddCommGroup H
inst✝⁸ : NormedSpace 𝕜✝ H
𝕝 : Type u_7
inst✝⁷ : NontriviallyNormedField 𝕝
inst✝⁶ : NormedAlgebra 𝕜✝ 𝕝
A✝ : Type u_8
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜✝ A✝
𝕜 : Type u_10
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_9
inst✝² : NormedRing A
inst✝¹ : NormOneClass A
inst✝ : NormedAlgebra 𝕜 A
r : ℝ≥0
hr : ↑r < FormalMultilinearSeries.radius (formalMultilinearSeries_geometric 𝕜 A)
⊢ r ≤ 1

Next tactic:
have := FormalMultilinearSeries.isLittleO_one_of_lt_radius _ hr
lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
    (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
    (formalMultilinearSeries_geometric 𝕜 A).radius = 1 := by
  apply le_antisymm
  · refine le_of_forall_nnreal_lt (fun r hr ↦ ?_)
    rw [← coe_one, ENNReal.coe_le_coe]
    have := FormalMultilinearSeries.isLittleO_one_of_lt_radius _ hr
    

Proof state:
case a
α : Type u_1
𝕜✝ : Type u_2
inst✝¹⁶ : NontriviallyNormedField 𝕜✝
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace 𝕜✝ E
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜✝ F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜✝ G
inst✝⁹ : NormedAddCommGroup H
inst✝⁸ : NormedSpace 𝕜✝ H
𝕝 : Type u_7
inst✝⁷ : NontriviallyNormedField 𝕝
inst✝⁶ : NormedAlgebra 𝕜✝ 𝕝
A✝ : Type u_8
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜✝ A✝
𝕜 : Type u_10
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_9
inst✝² : NormedRing A
inst✝¹ : NormOneClass A
inst✝ : NormedAlgebra 𝕜 A
r : ℝ≥0
hr : ↑r < FormalMultilinearSeries.radius (formalMultilinearSeries_geometric 𝕜 A)
this : (fun n => ‖formalMultilinearSeries_geometric 𝕜 A n‖ * ↑r ^ n) =o[atTop] fun x => 1
⊢ r ≤ 1

Next tactic:
simp_rw [formalMultilinearSeries_geometric_apply_norm, one_mul] at this
lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
    (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
    (formalMultilinearSeries_geometric 𝕜 A).radius = 1 := by
  apply le_antisymm
  · refine le_of_forall_nnreal_lt (fun r hr ↦ ?_)
    rw [← coe_one, ENNReal.coe_le_coe]
    have := FormalMultilinearSeries.isLittleO_one_of_lt_radius _ hr
    simp_rw [formalMultilinearSeries_geometric_apply_norm, one_mul] at this
    

Proof state:
case a
α : Type u_1
𝕜✝ : Type u_2
inst✝¹⁶ : NontriviallyNormedField 𝕜✝
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace 𝕜✝ E
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜✝ F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜✝ G
inst✝⁹ : NormedAddCommGroup H
inst✝⁸ : NormedSpace 𝕜✝ H
𝕝 : Type u_7
inst✝⁷ : NontriviallyNormedField 𝕝
inst✝⁶ : NormedAlgebra 𝕜✝ 𝕝
A✝ : Type u_8
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜✝ A✝
𝕜 : Type u_10
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_9
inst✝² : NormedRing A
inst✝¹ : NormOneClass A
inst✝ : NormedAlgebra 𝕜 A
r : ℝ≥0
hr : ↑r < FormalMultilinearSeries.radius (formalMultilinearSeries_geometric 𝕜 A)
this : (fun n => ↑r ^ n) =o[atTop] fun x => 1
⊢ r ≤ 1

Next tactic:
contrapose! this
lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
    (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
    (formalMultilinearSeries_geometric 𝕜 A).radius = 1 := by
  apply le_antisymm
  · refine le_of_forall_nnreal_lt (fun r hr ↦ ?_)
    rw [← coe_one, ENNReal.coe_le_coe]
    have := FormalMultilinearSeries.isLittleO_one_of_lt_radius _ hr
    simp_rw [formalMultilinearSeries_geometric_apply_norm, one_mul] at this
    contrapose! this
    

Proof state:
case a
α : Type u_1
𝕜✝ : Type u_2
inst✝¹⁶ : NontriviallyNormedField 𝕜✝
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace 𝕜✝ E
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜✝ F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜✝ G
inst✝⁹ : NormedAddCommGroup H
inst✝⁸ : NormedSpace 𝕜✝ H
𝕝 : Type u_7
inst✝⁷ : NontriviallyNormedField 𝕝
inst✝⁶ : NormedAlgebra 𝕜✝ 𝕝
A✝ : Type u_8
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜✝ A✝
𝕜 : Type u_10
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_9
inst✝² : NormedRing A
inst✝¹ : NormOneClass A
inst✝ : NormedAlgebra 𝕜 A
r : ℝ≥0
hr : ↑r < FormalMultilinearSeries.radius (formalMultilinearSeries_geometric 𝕜 A)
this : 1 < r
⊢ ¬(fun n => ↑r ^ n) =o[atTop] fun x => 1

Next tactic:
simp_rw [IsLittleO, IsBigOWith, not_forall, norm_one, mul_one,
      not_eventually]
lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
    (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
    (formalMultilinearSeries_geometric 𝕜 A).radius = 1 := by
  apply le_antisymm
  · refine le_of_forall_nnreal_lt (fun r hr ↦ ?_)
    rw [← coe_one, ENNReal.coe_le_coe]
    have := FormalMultilinearSeries.isLittleO_one_of_lt_radius _ hr
    simp_rw [formalMultilinearSeries_geometric_apply_norm, one_mul] at this
    contrapose! this
    simp_rw [IsLittleO, IsBigOWith, not_forall, norm_one, mul_one,
      not_eventually]
    

Proof state:
case a
α : Type u_1
𝕜✝ : Type u_2
inst✝¹⁶ : NontriviallyNormedField 𝕜✝
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace 𝕜✝ E
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜✝ F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜✝ G
inst✝⁹ : NormedAddCommGroup H
inst✝⁸ : NormedSpace 𝕜✝ H
𝕝 : Type u_7
inst✝⁷ : NontriviallyNormedField 𝕝
inst✝⁶ : NormedAlgebra 𝕜✝ 𝕝
A✝ : Type u_8
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜✝ A✝
𝕜 : Type u_10
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_9
inst✝² : NormedRing A
inst✝¹ : NormOneClass A
inst✝ : NormedAlgebra 𝕜 A
r : ℝ≥0
hr : ↑r < FormalMultilinearSeries.radius (formalMultilinearSeries_geometric 𝕜 A)
this : 1 < r
⊢ ∃ x, ∃ (_ : 0 < x), ∃ᶠ (x_1 : ℕ) in atTop, ¬‖↑r ^ x_1‖ ≤ x

Next tactic:
refine ⟨1, one_pos, ?_⟩
lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
    (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
    (formalMultilinearSeries_geometric 𝕜 A).radius = 1 := by
  apply le_antisymm
  · refine le_of_forall_nnreal_lt (fun r hr ↦ ?_)
    rw [← coe_one, ENNReal.coe_le_coe]
    have := FormalMultilinearSeries.isLittleO_one_of_lt_radius _ hr
    simp_rw [formalMultilinearSeries_geometric_apply_norm, one_mul] at this
    contrapose! this
    simp_rw [IsLittleO, IsBigOWith, not_forall, norm_one, mul_one,
      not_eventually]
    refine ⟨1, one_pos, ?_⟩
    

Proof state:
case a
α : Type u_1
𝕜✝ : Type u_2
inst✝¹⁶ : NontriviallyNormedField 𝕜✝
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace 𝕜✝ E
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜✝ F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜✝ G
inst✝⁹ : NormedAddCommGroup H
inst✝⁸ : NormedSpace 𝕜✝ H
𝕝 : Type u_7
inst✝⁷ : NontriviallyNormedField 𝕝
inst✝⁶ : NormedAlgebra 𝕜✝ 𝕝
A✝ : Type u_8
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜✝ A✝
𝕜 : Type u_10
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_9
inst✝² : NormedRing A
inst✝¹ : NormOneClass A
inst✝ : NormedAlgebra 𝕜 A
r : ℝ≥0
hr : ↑r < FormalMultilinearSeries.radius (formalMultilinearSeries_geometric 𝕜 A)
this : 1 < r
⊢ ∃ᶠ (x : ℕ) in atTop, ¬‖↑r ^ x‖ ≤ 1

Next tactic:
refine ((eventually_ne_atTop 0).mp (eventually_of_forall ?_)).frequently
lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
    (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
    (formalMultilinearSeries_geometric 𝕜 A).radius = 1 := by
  apply le_antisymm
  · refine le_of_forall_nnreal_lt (fun r hr ↦ ?_)
    rw [← coe_one, ENNReal.coe_le_coe]
    have := FormalMultilinearSeries.isLittleO_one_of_lt_radius _ hr
    simp_rw [formalMultilinearSeries_geometric_apply_norm, one_mul] at this
    contrapose! this
    simp_rw [IsLittleO, IsBigOWith, not_forall, norm_one, mul_one,
      not_eventually]
    refine ⟨1, one_pos, ?_⟩
    refine ((eventually_ne_atTop 0).mp (eventually_of_forall ?_)).frequently
    

Proof state:
case a
α : Type u_1
𝕜✝ : Type u_2
inst✝¹⁶ : NontriviallyNormedField 𝕜✝
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace 𝕜✝ E
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜✝ F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜✝ G
inst✝⁹ : NormedAddCommGroup H
inst✝⁸ : NormedSpace 𝕜✝ H
𝕝 : Type u_7
inst✝⁷ : NontriviallyNormedField 𝕝
inst✝⁶ : NormedAlgebra 𝕜✝ 𝕝
A✝ : Type u_8
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜✝ A✝
𝕜 : Type u_10
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_9
inst✝² : NormedRing A
inst✝¹ : NormOneClass A
inst✝ : NormedAlgebra 𝕜 A
r : ℝ≥0
hr : ↑r < FormalMultilinearSeries.radius (formalMultilinearSeries_geometric 𝕜 A)
this : 1 < r
⊢ ∀ (x : ℕ), x ≠ 0 → ¬‖↑r ^ x‖ ≤ 1

Next tactic:
intro n hn
lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
    (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
    (formalMultilinearSeries_geometric 𝕜 A).radius = 1 := by
  apply le_antisymm
  · refine le_of_forall_nnreal_lt (fun r hr ↦ ?_)
    rw [← coe_one, ENNReal.coe_le_coe]
    have := FormalMultilinearSeries.isLittleO_one_of_lt_radius _ hr
    simp_rw [formalMultilinearSeries_geometric_apply_norm, one_mul] at this
    contrapose! this
    simp_rw [IsLittleO, IsBigOWith, not_forall, norm_one, mul_one,
      not_eventually]
    refine ⟨1, one_pos, ?_⟩
    refine ((eventually_ne_atTop 0).mp (eventually_of_forall ?_)).frequently
    intro n hn
    

Proof state:
case a
α : Type u_1
𝕜✝ : Type u_2
inst✝¹⁶ : NontriviallyNormedField 𝕜✝
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedSpace 𝕜✝ E
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜✝ F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NormedSpace 𝕜✝ G
inst✝⁹ : NormedAddCommGroup H
inst✝⁸ : NormedSpace 𝕜✝ H
𝕝 : Type u_7
inst✝⁷ : NontriviallyNormedField 𝕝
inst✝⁶ : NormedAlgebra 𝕜✝ 𝕝
A✝ : Type u_8
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜✝ A✝
𝕜 : Type u_10
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_9
inst✝² : NormedRing A
inst✝¹ : NormOneClass A
inst✝ : NormedAlgebra 𝕜 A
r : ℝ≥0
hr : ↑r < FormalMultilinearSeries.radius (formalMultilinearSeries_geometric 𝕜 A)
this : 1 < r
n : ℕ
hn : n ≠ 0
⊢ ¬‖↑r ^ n‖ ≤ 1

Next tactic:
push_neg