awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
⊢ I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k =
    I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n))
have rw : r * S n ≠ 1
Tactic: rw [geom_sum_Ico' _ In]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
⊢ I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) ≤
    I * a + I * C * (HPow.hPow (β := ℕ) (r * S n) 2 / (1 / 2))
have bc : HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1) ≤ HPow.hPow (β := ℕ) (r * S n) 2
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
bc : HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1) ≤ HPow.hPow (β := ℕ) (r * S n) 2
⊢ I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) ≤
    I * a + I * C * (HPow.hPow (β := ℕ) (r * S n) 2 / (1 / 2))
have bc₁ : 1 / 2 ≤ 1 - r * S n
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
⊢ HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1) ≤ HPow.hPow (β := ℕ) (r * S n) 2
have bc : 0 ≤ (r * S n) ^ (n + 1)
Tactic: simp only [sub_le_self_iff]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
⊢ 1 / 2 ≤ 1 - r * S n
have linarith : 1 - r * S n < 1 / 2 → HMul.hMul (α := ℝ) 1 2 = 2
Tactic: linarith only [rSn]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
linarith : 1 - r * S n < 1 / 2 → HMul.hMul (α := ℝ) 1 2 = 2
⊢ 1 / 2 ≤ 1 - r * S n
have linarith₁ : 1 - r * S n < 1 / 2 → 2 / 2 = 1
Tactic: linarith only [rSn]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
linarith : 1 - r * S n < 1 / 2 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₁ : 1 - r * S n < 1 / 2 → 2 / 2 = 1
linarith₂ : 1 - r * S n < 1 / 2 → HMul.hMul (α := ℝ) 1 2 = 2
⊢ 1 / 2 ≤ 1 - r * S n
have linarith₃ : 1 - r * S n < 1 / 2 → GT.gt (α := ℝ) 2 0
Tactic: linarith only [rSn]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
linarith : 1 - r * S n < 1 / 2 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₁ : 1 - r * S n < 1 / 2 → 2 / 2 = 1
linarith₂ : 1 - r * S n < 1 / 2 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₃ : 1 - r * S n < 1 / 2 → GT.gt (α := ℝ) 2 0
linarith₄ : 1 - r * S n < 1 / 2 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₅ : 1 - r * S n < 1 / 2 → 2 / 2 = 1
linarith₆ : 1 - r * S n < 1 / 2 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₇ : 1 - r * S n < 1 / 2 → GT.gt (α := ℝ) 2 0
⊢ 1 / 2 ≤ 1 - r * S n
have linarith₈ : 1 - r * S n < 1 / 2 →
  HMul.hMul (α := ℝ) 2 1 - 1 * r * (2 * S n) - HMul.hMul (α := ℝ) 1 1 + (1 * r * (2 * S n) - HMul.hMul (α := ℝ) 1 1) = 0
Tactic: linarith only [rSn]
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
⊢ I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * S n) 2 ≤ I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * ((I + 1) * a)) 2
have bc : 0 ≤ 2 * I * C
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
bc : 0 ≤ 2 * I * C
⊢ I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * S n) 2 ≤ I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * ((I + 1) * a)) 2
have bc₁ : 0 ≤ r * S n
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
bc : 0 ≤ 2 * I * C
bc₁ : 0 ≤ r * S n
⊢ I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * S n) 2 ≤ I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * ((I + 1) * a)) 2
have bc₂ : 0 ≤ r
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
⊢ (I + 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a) * a ≤ (I + 1) * a
have ha : 0 ≤ a
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
⊢ S (n + 1) ≤ (I + 1) * a
have S₁ : I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k =
  I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n))
Tactic: calc
        S (n + 1) ≤ I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k :=
          radius_rightInv_pos_of_radius_pos_aux2 In p i rpos.le apos.le Cpos.le ple
        _ = I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) := by
          rw [geom_sum_Ico' _ In]; exact ne_of_lt (rSn.trans_lt (by norm_num))
        _ ≤ I * a + I * C * ((r * S n) ^ 2 / (1 / 2)) := by
          gcongr
          · simp only [sub_le_self_iff]
            positivity
          · linarith only [rSn]
        _ = I * a + 2 * I * C * (r * S n) ^ 2 := by ring
        _ ≤ I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 := by gcongr
        _ = (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a := by ring
        _ ≤ (I + 1) * a := by gcongr
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
S₁ :
  I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k =
    I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n))
⊢ S (n + 1) ≤ (I + 1) * a
have S₂ : I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) ≤
  I * a + I * C * (HPow.hPow (β := ℕ) (r * S n) 2 / (1 / 2))
Tactic: calc
        S (n + 1) ≤ I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k :=
          radius_rightInv_pos_of_radius_pos_aux2 In p i rpos.le apos.le Cpos.le ple
        _ = I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) := by
          rw [geom_sum_Ico' _ In]; exact ne_of_lt (rSn.trans_lt (by norm_num))
        _ ≤ I * a + I * C * ((r * S n) ^ 2 / (1 / 2)) := by
          gcongr
          · simp only [sub_le_self_iff]
            positivity
          · linarith only [rSn]
        _ = I * a + 2 * I * C * (r * S n) ^ 2 := by ring
        _ ≤ I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 := by gcongr
        _ = (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a := by ring
        _ ≤ (I + 1) * a := by gcongr
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
S₁ :
  I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k =
    I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n))
S₂ :
  I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) ≤
    I * a + I * C * (HPow.hPow (β := ℕ) (r * S n) 2 / (1 / 2))
⊢ S (n + 1) ≤ (I + 1) * a
have S₃ : I * a + I * C * (HPow.hPow (β := ℕ) (r * S n) 2 / (1 / 2)) = I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * S n) 2
Tactic: calc
        S (n + 1) ≤ I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k :=
          radius_rightInv_pos_of_radius_pos_aux2 In p i rpos.le apos.le Cpos.le ple
        _ = I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) := by
          rw [geom_sum_Ico' _ In]; exact ne_of_lt (rSn.trans_lt (by norm_num))
        _ ≤ I * a + I * C * ((r * S n) ^ 2 / (1 / 2)) := by
          gcongr
          · simp only [sub_le_self_iff]
            positivity
          · linarith only [rSn]
        _ = I * a + 2 * I * C * (r * S n) ^ 2 := by ring
        _ ≤ I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 := by gcongr
        _ = (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a := by ring
        _ ≤ (I + 1) * a := by gcongr
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
S₁ :
  I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k =
    I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n))
S₂ :
  I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) ≤
    I * a + I * C * (HPow.hPow (β := ℕ) (r * S n) 2 / (1 / 2))
S₃ : I * a + I * C * (HPow.hPow (β := ℕ) (r * S n) 2 / (1 / 2)) = I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * S n) 2
⊢ S (n + 1) ≤ (I + 1) * a
have S₄ : I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * S n) 2 ≤ I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * ((I + 1) * a)) 2
Tactic: calc
        S (n + 1) ≤ I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k :=
          radius_rightInv_pos_of_radius_pos_aux2 In p i rpos.le apos.le Cpos.le ple
        _ = I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) := by
          rw [geom_sum_Ico' _ In]; exact ne_of_lt (rSn.trans_lt (by norm_num))
        _ ≤ I * a + I * C * ((r * S n) ^ 2 / (1 / 2)) := by
          gcongr
          · simp only [sub_le_self_iff]
            positivity
          · linarith only [rSn]
        _ = I * a + 2 * I * C * (r * S n) ^ 2 := by ring
        _ ≤ I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 := by gcongr
        _ = (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a := by ring
        _ ≤ (I + 1) * a := by gcongr
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
S₁ :
  I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k =
    I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n))
S₂ :
  I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) ≤
    I * a + I * C * (HPow.hPow (β := ℕ) (r * S n) 2 / (1 / 2))
S₃ : I * a + I * C * (HPow.hPow (β := ℕ) (r * S n) 2 / (1 / 2)) = I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * S n) 2
S₄ : I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * S n) 2 ≤ I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * ((I + 1) * a)) 2
⊢ S (n + 1) ≤ (I + 1) * a
have S₅ : I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * ((I + 1) * a)) 2 =
  (I + 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a) * a
Tactic: calc
        S (n + 1) ≤ I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k :=
          radius_rightInv_pos_of_radius_pos_aux2 In p i rpos.le apos.le Cpos.le ple
        _ = I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) := by
          rw [geom_sum_Ico' _ In]; exact ne_of_lt (rSn.trans_lt (by norm_num))
        _ ≤ I * a + I * C * ((r * S n) ^ 2 / (1 / 2)) := by
          gcongr
          · simp only [sub_le_self_iff]
            positivity
          · linarith only [rSn]
        _ = I * a + 2 * I * C * (r * S n) ^ 2 := by ring
        _ ≤ I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 := by gcongr
        _ = (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a := by ring
        _ ≤ (I + 1) * a := by gcongr
-- Lean 4 proof step from Mathlib.Analysis.Analytic.Inverse (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖(↑i.symm : F →L[𝕜] E)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : (n : ℕ) → ℝ := fun (n : ℕ) ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
S₁ :
  I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k =
    I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n))
S₂ :
  I * a + I * C * ((HPow.hPow (β := ℕ) (r * S n) 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) ≤
    I * a + I * C * (HPow.hPow (β := ℕ) (r * S n) 2 / (1 / 2))
S₃ : I * a + I * C * (HPow.hPow (β := ℕ) (r * S n) 2 / (1 / 2)) = I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * S n) 2
S₄ : I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * S n) 2 ≤ I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * ((I + 1) * a)) 2
S₅ :
  I * a + 2 * I * C * HPow.hPow (β := ℕ) (r * ((I + 1) * a)) 2 =
    (I + 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a) * a
⊢ S (n + 1) ≤ (I + 1) * a
have S₆ : (I + 2 * I * C * HPow.hPow (β := ℕ) r 2 * HPow.hPow (β := ℕ) (I + 1) 2 * a) * a ≤ (I + 1) * a
Tactic: calc
        S (n + 1) ≤ I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k :=
          radius_rightInv_pos_of_radius_pos_aux2 In p i rpos.le apos.le Cpos.le ple
        _ = I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) := by
          rw [geom_sum_Ico' _ In]; exact ne_of_lt (rSn.trans_lt (by norm_num))
        _ ≤ I * a + I * C * ((r * S n) ^ 2 / (1 / 2)) := by
          gcongr
          · simp only [sub_le_self_iff]
            positivity
          · linarith only [rSn]
        _ = I * a + 2 * I * C * (r * S n) ^ 2 := by ring
        _ ≤ I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 := by gcongr
        _ = (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a := by ring
        _ ≤ (I + 1) * a := by gcongr