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train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	crucial	[7, 1]	[49, 57]	simp [h4, deriv_mul e1 e2, this]	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z this : deriv (fun y => y - z₀) z = 1 ⊢ deriv f z = g z + (z - z₀) * deriv g z	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	crucial	[7, 1]	[49, 57]	change deriv (fun y => id y - z₀) z = 1	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z ⊢ deriv (fun y => y - z₀) z = 1	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z ⊢ deriv (fun y => id y - z₀) z = 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	crucial	[7, 1]	[49, 57]	simp [deriv_sub_const]	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z ⊢ deriv (fun y => id y - z₀) z = 1	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	crucial	[7, 1]	[49, 57]	rw [circleIntegrable_sub_inv_iff, abs_eq_self.2 hr.le]	case e_a.hf ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ CircleIntegrable (fun z => (z - z₀)⁻¹) c r	case e_a.hf ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ r = 0 ∨ z₀ ∉ sphere c r
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	crucial	[7, 1]	[49, 57]	exact Or.inr (disjoint_right.1 sphere_disjoint_ball hz₀)	case e_a.hf ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ r = 0 ∨ z₀ ∉ sphere c r	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	crucial	[7, 1]	[49, 57]	refine (ContinuousOn.div ?_ ?_ ?_).circleIntegrable hr.le	case e_a.hg ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ CircleIntegrable (fun z => deriv g z / g z) c r	case e_a.hg.refine_1 ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ ContinuousOn (fun z => deriv g z) (sphere c r) case e_a.hg.refine_2 ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ ContinuousOn (fun z => g z) (sphere c r) case e_a.hg.refine_3 ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ ∀ x ∈ sphere c r, g x ≠ 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	crucial	[7, 1]	[49, 57]	exact (h1.deriv hU).continuousOn.mono (sphere_subset_closedBall.trans hcr)	case e_a.hg.refine_1 ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ ContinuousOn (fun z => deriv g z) (sphere c r)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	crucial	[7, 1]	[49, 57]	exact h1.continuousOn.mono (sphere_subset_closedBall.trans hcr)	case e_a.hg.refine_2 ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ ContinuousOn (fun z => g z) (sphere c r)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	crucial	[7, 1]	[49, 57]	exact λ z hz => h2 z (sphere_subset_closedBall hz)	case e_a.hg.refine_3 ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ : ℂ → ℂ hU : IsOpen U hcr : closedBall c r ⊆ U hz₀ : z₀ ∈ ball c r hf : DifferentiableOn ℂ f U hfz₀ : f z₀ = 0 hf'z₀ : deriv f z₀ ≠ 0 hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0 hr : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ ∀ x ∈ sphere c r, g x ≠ 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	bla	[55, 1]	[67, 95]	have h1 : ∀ᶠ z in 𝓝 z₀, AnalyticAt ℂ f z := (isOpen_analyticAt ℂ f).mem_nhds hf	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ ⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	bla	[55, 1]	[67, 95]	obtain ⟨ε, hε, h⟩ := Metric.mem_nhds_iff.1 (h1.and hf'.eventually_eq_zero)	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀	case intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} ⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	bla	[55, 1]	[67, 95]	refine Metric.mem_nhds_iff.2 ⟨ε, hε, λ z hz => ?_⟩	case intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} ⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀	case intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z : ℂ hz : z ∈ ball z₀ ε ⊢ z ∈ {x \| (fun z => f z = f z₀) x}
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	bla	[55, 1]	[67, 95]	have h3 : ∀ z ∈ ball z₀ ε, fderivWithin ℂ f (ball z₀ ε) z = 0 := by rintro z hz rw [fderivWithin_eq_fderiv (isOpen_ball.uniqueDiffWithinAt hz) ((h hz).1.differentiableAt)] ext1 simpa [fderiv_deriv] using (h hz).2	case intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z : ℂ hz : z ∈ ball z₀ ε ⊢ z ∈ {x \| (fun z => f z = f z₀) x}	case intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z : ℂ hz : z ∈ ball z₀ ε h3 : ∀ z ∈ ball z₀ ε, fderivWithin ℂ f (ball z₀ ε) z = 0 ⊢ z ∈ {x \| (fun z => f z = f z₀) x}
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	bla	[55, 1]	[67, 95]	have h4 : DifferentiableOn ℂ f (ball z₀ ε) := λ z hz => (h hz).1.differentiableWithinAt	case intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z : ℂ hz : z ∈ ball z₀ ε h3 : ∀ z ∈ ball z₀ ε, fderivWithin ℂ f (ball z₀ ε) z = 0 ⊢ z ∈ {x \| (fun z => f z = f z₀) x}	case intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z : ℂ hz : z ∈ ball z₀ ε h3 : ∀ z ∈ ball z₀ ε, fderivWithin ℂ f (ball z₀ ε) z = 0 h4 : DifferentiableOn ℂ f (ball z₀ ε) ⊢ z ∈ {x \| (fun z => f z = f z₀) x}
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	bla	[55, 1]	[67, 95]	exact Convex.is_const_of_fderivWithin_eq_zero (convex_ball z₀ ε) h4 h3 hz (mem_ball_self hε)	case intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z : ℂ hz : z ∈ ball z₀ ε h3 : ∀ z ∈ ball z₀ ε, fderivWithin ℂ f (ball z₀ ε) z = 0 h4 : DifferentiableOn ℂ f (ball z₀ ε) ⊢ z ∈ {x \| (fun z => f z = f z₀) x}	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	bla	[55, 1]	[67, 95]	rintro z hz	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z : ℂ hz : z ∈ ball z₀ ε ⊢ ∀ z ∈ ball z₀ ε, fderivWithin ℂ f (ball z₀ ε) z = 0	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z✝ : ℂ hz✝ : z✝ ∈ ball z₀ ε z : ℂ hz : z ∈ ball z₀ ε ⊢ fderivWithin ℂ f (ball z₀ ε) z = 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	bla	[55, 1]	[67, 95]	rw [fderivWithin_eq_fderiv (isOpen_ball.uniqueDiffWithinAt hz) ((h hz).1.differentiableAt)]	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z✝ : ℂ hz✝ : z✝ ∈ ball z₀ ε z : ℂ hz : z ∈ ball z₀ ε ⊢ fderivWithin ℂ f (ball z₀ ε) z = 0	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z✝ : ℂ hz✝ : z✝ ∈ ball z₀ ε z : ℂ hz : z ∈ ball z₀ ε ⊢ fderiv ℂ f z = 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	bla	[55, 1]	[67, 95]	ext1	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z✝ : ℂ hz✝ : z✝ ∈ ball z₀ ε z : ℂ hz : z ∈ ball z₀ ε ⊢ fderiv ℂ f z = 0	case h ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z✝ : ℂ hz✝ : z✝ ∈ ball z₀ ε z : ℂ hz : z ∈ ball z₀ ε ⊢ (fderiv ℂ f z) 1 = 0 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	bla	[55, 1]	[67, 95]	simpa [fderiv_deriv] using (h hz).2	case h ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ hf : AnalyticAt ℂ f z₀ hf' : HasFPowerSeriesAt (deriv f) 0 z₀ h1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z ε : ℝ hε : ε > 0 h : ball z₀ ε ⊆ {x \| (fun x => AnalyticAt ℂ f x ∧ deriv f x = 0) x} z✝ : ℂ hz✝ : z✝ ∈ ball z₀ ε z : ℂ hz : z ∈ ball z₀ ε ⊢ (fderiv ℂ f z) 1 = 0 1	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	classical have h1 : p.coeff 1 = 0 := by simpa only [hg'] using hgp.deriv.symm have h2 : p 0 = 0 := by ext1 x; simpa only [hg] using hgp.coeff_zero x have h3 : p 1 = 0 := by ext1; simp [h1] rw [FormalMultilinearSeries.order_eq_find' hp, Nat.le_find_iff] intro n hn cases n case zero => simp [h2] case succ n => cases n case zero => simpa using h3 case succ => linarith	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 ⊢ 2 ≤ FormalMultilinearSeries.order p	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	have h1 : p.coeff 1 = 0 := by simpa only [hg'] using hgp.deriv.symm	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 ⊢ 2 ≤ FormalMultilinearSeries.order p	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 ⊢ 2 ≤ FormalMultilinearSeries.order p
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	have h2 : p 0 = 0 := by ext1 x; simpa only [hg] using hgp.coeff_zero x	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 ⊢ 2 ≤ FormalMultilinearSeries.order p	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 ⊢ 2 ≤ FormalMultilinearSeries.order p
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	have h3 : p 1 = 0 := by ext1; simp [h1]	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 ⊢ 2 ≤ FormalMultilinearSeries.order p	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 ⊢ 2 ≤ FormalMultilinearSeries.order p
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	rw [FormalMultilinearSeries.order_eq_find' hp, Nat.le_find_iff]	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 ⊢ 2 ≤ FormalMultilinearSeries.order p	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 ⊢ ∀ m < 2, ¬p m ≠ 0 m
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	intro n hn	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 ⊢ ∀ m < 2, ¬p m ≠ 0 m	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 n : ℕ hn : n < 2 ⊢ ¬p n ≠ 0 n
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	cases n	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 n : ℕ hn : n < 2 ⊢ ¬p n ≠ 0 n	case zero ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 hn : Nat.zero < 2 ⊢ ¬p Nat.zero ≠ 0 Nat.zero case succ ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 n✝ : ℕ hn : Nat.succ n✝ < 2 ⊢ ¬p (Nat.succ n✝) ≠ 0 (Nat.succ n✝)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	case zero => simp [h2]	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 hn : Nat.zero < 2 ⊢ ¬p Nat.zero ≠ 0 Nat.zero	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	case succ n => cases n case zero => simpa using h3 case succ => linarith	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 n : ℕ hn : Nat.succ n < 2 ⊢ ¬p (Nat.succ n) ≠ 0 (Nat.succ n)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	simpa only [hg'] using hgp.deriv.symm	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 ⊢ FormalMultilinearSeries.coeff p 1 = 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	ext1 x	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 ⊢ p 0 = 0	case H ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 x : Fin 0 → ℂ ⊢ (p 0) x = 0 x
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	simpa only [hg] using hgp.coeff_zero x	case H ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 x : Fin 0 → ℂ ⊢ (p 0) x = 0 x	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	ext1	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 ⊢ p 1 = 0	case H ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 x✝ : Fin 1 → ℂ ⊢ (p 1) x✝ = 0 x✝
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	simp [h1]	case H ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 x✝ : Fin 1 → ℂ ⊢ (p 1) x✝ = 0 x✝	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	simp [h2]	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 hn : Nat.zero < 2 ⊢ ¬p Nat.zero ≠ 0 Nat.zero	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	cases n	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 n : ℕ hn : Nat.succ n < 2 ⊢ ¬p (Nat.succ n) ≠ 0 (Nat.succ n)	case zero ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 hn : Nat.succ Nat.zero < 2 ⊢ ¬p (Nat.succ Nat.zero) ≠ 0 (Nat.succ Nat.zero) case succ ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 n✝ : ℕ hn : Nat.succ (Nat.succ n✝) < 2 ⊢ ¬p (Nat.succ (Nat.succ n✝)) ≠ 0 (Nat.succ (Nat.succ n✝))
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	case zero => simpa using h3	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 hn : Nat.succ Nat.zero < 2 ⊢ ¬p (Nat.succ Nat.zero) ≠ 0 (Nat.succ Nat.zero)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	case succ => linarith	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 n✝ : ℕ hn : Nat.succ (Nat.succ n✝) < 2 ⊢ ¬p (Nat.succ (Nat.succ n✝)) ≠ 0 (Nat.succ (Nat.succ n✝))	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	simpa using h3	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 hn : Nat.succ Nat.zero < 2 ⊢ ¬p (Nat.succ Nat.zero) ≠ 0 (Nat.succ Nat.zero)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	two_le_order_of_deriv_eq_zero	[69, 1]	[83, 26]	linarith	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ p : FormalMultilinearSeries ℂ ℂ ℂ hgp : HasFPowerSeriesAt g p z₀ hp : p ≠ 0 hg : g z₀ = 0 hg' : deriv g z₀ = 0 h1 : FormalMultilinearSeries.coeff p 1 = 0 h2 : p 0 = 0 h3 : p 1 = 0 n✝ : ℕ hn : Nat.succ (Nat.succ n✝) < 2 ⊢ ¬p (Nat.succ (Nat.succ n✝)) ≠ 0 (Nat.succ (Nat.succ n✝))	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	tendsto_uniformly_on_add_const	[85, 1]	[89, 50]	have : Tendsto id (𝓝[≠] (0 : ℂ)) (𝓝 0) := nhdsWithin_le_nhds	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ ⊢ TendstoUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ this : Tendsto id (𝓝[≠] 0) (𝓝 0) ⊢ TendstoUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	tendsto_uniformly_on_add_const	[85, 1]	[89, 50]	have : TendstoUniformlyOn (λ (ε _ : ℂ) => ε) 0 (𝓝[≠] 0) U := this.tendstoUniformlyOn_const U	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ this : Tendsto id (𝓝[≠] 0) (𝓝 0) ⊢ TendstoUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ this✝ : Tendsto id (𝓝[≠] 0) (𝓝 0) this : TendstoUniformlyOn (fun ε x => ε) 0 (𝓝[≠] 0) U ⊢ TendstoUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	tendsto_uniformly_on_add_const	[85, 1]	[89, 50]	simpa using tendsto_uniformly_on_const.add this	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g : ℂ → ℂ this✝ : Tendsto id (𝓝[≠] 0) (𝓝 0) this : TendstoUniformlyOn (fun ε x => ε) 0 (𝓝[≠] 0) U ⊢ TendstoUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	obtain ⟨p, hp⟩ : AnalyticAt ℂ g z₀ := hg.analyticAt (hU.mem_nhds hz₀)	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 ⊢ deriv g z₀ ≠ 0	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ ⊢ deriv g z₀ ≠ 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have h25 : ∀ᶠ z in 𝓝[≠] z₀, g z ≠ 0 := by simp only [eventually_nhdsWithin_iff] filter_upwards [hU.eventually_mem hz₀] with z hz hzz₀ simpa only [hgz₀] using hi.ne hz hz₀ hzz₀	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ ⊢ deriv g z₀ ≠ 0	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 ⊢ deriv g z₀ ≠ 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have h17 : p ≠ 0 := by simpa [← hp.locally_zero_iff.not] using h25.frequently.filter_mono nhdsWithin_le_nhds	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 ⊢ deriv g z₀ ≠ 0	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 ⊢ deriv g z₀ ≠ 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	by_contra h	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 ⊢ deriv g z₀ ≠ 0	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have h6 : 2 ≤ p.order := two_le_order_of_deriv_eq_zero hp h17 hgz₀ h	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 ⊢ False	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	obtain ⟨r, h7, h8, h14, h21, h20⟩ : ∃ r > 0, cindex z₀ r g = p.order ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p ⊢ False	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 := λ z hz => h21 z (sphere_subset_closedBall hz) (ne_of_mem_sphere hz h7.lt.ne.symm)	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U ⊢ False	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have h18 : ∀ ε, DifferentiableOn ℂ (λ z => g z + ε) U := λ ε => hg.add_const ε	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 ⊢ False	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have h19 : TendstoLocallyUniformlyOn (λ ε z => g z + ε) g (𝓝[≠] 0) U := tendsto_uniformly_on_add_const.tendstoLocallyUniformlyOn	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U ⊢ False	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have h9 : ∀ᶠ ε in 𝓝[≠] 0, cindex z₀ r (λ z => g z + ε) = 1 := by have h24 : p.order ≠ 0 := by linarith have := hurwitz2 hU (eventually_of_forall h18) h19 h7 h20 h22 (by simp [h8, h24]) simp only [eventually_nhdsWithin_iff] at this ⊢ filter_upwards [this] with ε h hε obtain ⟨z, hz, hgz⟩ := h hε have e1 : z ≠ z₀ := by rintro rfl; rw [hgz₀, zero_add] at hgz; exact hε hgz have e2 : deriv (λ z => g z + ε) z ≠ 0 := by simpa using h14 z (ball_subset_closedBall hz) e1 refine crucial hU h20 hz (h18 ε) hgz e2 (λ w hw hwz => ?_) contrapose! hwz exact hi (h20 hw) ((ball_subset_closedBall.trans h20) hz) (add_right_cancel (hwz.trans hgz.symm))	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U ⊢ False	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h9 : ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1 ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have h10 : Tendsto (λ ε => cindex z₀ r (λ z => g z + ε)) (𝓝[≠] 0) (𝓝 (cindex z₀ r g)) := hurwitz2_2 hU (eventually_of_forall h18) h19 h7 (sphere_subset_closedBall.trans h20) h22	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h9 : ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1 ⊢ False	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h9 : ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1 h10 : Tendsto (fun ε => cindex z₀ r fun z => g z + ε) (𝓝[≠] 0) (𝓝 (cindex z₀ r g)) ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	rw [tendsto_nhds_unique (Tendsto.congr' h9 h10) tendsto_const_nhds] at h8	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h9 : ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1 h10 : Tendsto (fun ε => cindex z₀ r fun z => g z + ε) (𝓝[≠] 0) (𝓝 (cindex z₀ r g)) ⊢ False	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : 1 = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h9 : ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1 h10 : Tendsto (fun ε => cindex z₀ r fun z => g z + ε) (𝓝[≠] 0) (𝓝 (cindex z₀ r g)) ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	norm_cast at h8	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : 1 = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h9 : ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1 h10 : Tendsto (fun ε => cindex z₀ r fun z => g z + ε) (𝓝[≠] 0) (𝓝 (cindex z₀ r g)) ⊢ False	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h9 : ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1 h10 : Tendsto (fun ε => cindex z₀ r fun z => g z + ε) (𝓝[≠] 0) (𝓝 (cindex z₀ r g)) h8 : 1 = FormalMultilinearSeries.order p ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	linarith	case intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h9 : ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1 h10 : Tendsto (fun ε => cindex z₀ r fun z => g z + ε) (𝓝[≠] 0) (𝓝 (cindex z₀ r g)) h8 : 1 = FormalMultilinearSeries.order p ⊢ False	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	simp only [eventually_nhdsWithin_iff]	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ ⊢ ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ ⊢ ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → g x ≠ 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	filter_upwards [hU.eventually_mem hz₀] with z hz hzz₀	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ ⊢ ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → g x ≠ 0	case h ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ z : ℂ hz : z ∈ U hzz₀ : z ∈ {z₀}ᶜ ⊢ g z ≠ 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	simpa only [hgz₀] using hi.ne hz hz₀ hzz₀	case h ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ z : ℂ hz : z ∈ U hzz₀ : z ∈ {z₀}ᶜ ⊢ g z ≠ 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	simpa [← hp.locally_zero_iff.not] using h25.frequently.filter_mono nhdsWithin_le_nhds	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 ⊢ p ≠ 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	obtain ⟨q, hq⟩ : AnalyticAt ℂ (deriv g) z₀ := (hg.deriv hU).analyticAt (hU.mem_nhds hz₀)	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have h26 : q ≠ 0 := by rintro rfl simpa [hgz₀] using (((bla ⟨p, hp⟩ hq).filter_mono nhdsWithin_le_nhds).and h25).exists	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have e1 := cindex_eventually_eq_order hp	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 e1 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have e2 := hp.locally_ne_zero h17	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 e1 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 e1 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) e2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have e3 := hq.locally_ne_zero h26	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 e1 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) e2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 e1 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) e2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 e3 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, deriv g z ≠ 0 ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have e4 := hU.eventually_mem hz₀	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 e1 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) e2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 e3 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, deriv g z ≠ 0 ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 e1 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) e2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 e3 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, deriv g z ≠ 0 e4 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ U ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	simp only [eventually_nhdsWithin_iff, mem_compl_singleton_iff] at e2 e3	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 e1 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) e2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 e3 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, deriv g z ≠ 0 e4 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ U ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 e1 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) e4 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ U e2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ≠ z₀ → g x ≠ 0 e3 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ≠ z₀ → deriv g x ≠ 0 ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	simp only [eventually_nhds_iff_eventually_closed_ball] at e2 e3 e4	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 e1 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) e4 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ U e2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ≠ z₀ → g x ≠ 0 e3 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ≠ z₀ → deriv g x ≠ 0 ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 e1 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) e2 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 e3 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 e4 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ z ∈ closedBall z₀ r, z ∈ U ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	exact (e1.and (e3.and (e2.and e4))).exists'	case intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ h26 : q ≠ 0 e1 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) e2 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 e3 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 e4 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ z ∈ closedBall z₀ r, z ∈ U ⊢ ∃ r > 0, cindex z₀ r g = ↑(FormalMultilinearSeries.order p) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0) ∧ (∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0) ∧ closedBall z₀ r ⊆ U	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	rintro rfl	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p q : FormalMultilinearSeries ℂ ℂ ℂ hq : HasFPowerSeriesAt (deriv g) q z₀ ⊢ q ≠ 0	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p hq : HasFPowerSeriesAt (deriv g) 0 z₀ ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	simpa [hgz₀] using (((bla ⟨p, hp⟩ hq).filter_mono nhdsWithin_le_nhds).and h25).exists	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p hq : HasFPowerSeriesAt (deriv g) 0 z₀ ⊢ False	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have h24 : p.order ≠ 0 := by linarith	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U ⊢ ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 ⊢ ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have := hurwitz2 hU (eventually_of_forall h18) h19 h7 h20 h22 (by simp [h8, h24])	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 ⊢ ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (n : ℂ) in 𝓝[≠] 0, ∃ z ∈ ball z₀ r, g z + n = 0 ⊢ ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	simp only [eventually_nhdsWithin_iff] at this ⊢	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (n : ℂ) in 𝓝[≠] 0, ∃ z ∈ ball z₀ r, g z + n = 0 ⊢ ∀ᶠ (ε : ℂ) in 𝓝[≠] 0, (cindex z₀ r fun z => g z + ε) = 1	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ⊢ ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → (cindex z₀ r fun z => g z + x) = 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	filter_upwards [this] with ε h hε	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ⊢ ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → (cindex z₀ r fun z => g z + x) = 1	case h ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ ⊢ (cindex z₀ r fun z => g z + ε) = 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	obtain ⟨z, hz, hgz⟩ := h hε	case h ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ ⊢ (cindex z₀ r fun z => g z + ε) = 1	case h.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ z : ℂ hz : z ∈ ball z₀ r hgz : g z + ε = 0 ⊢ (cindex z₀ r fun z => g z + ε) = 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have e1 : z ≠ z₀ := by rintro rfl; rw [hgz₀, zero_add] at hgz; exact hε hgz	case h.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ z : ℂ hz : z ∈ ball z₀ r hgz : g z + ε = 0 ⊢ (cindex z₀ r fun z => g z + ε) = 1	case h.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ z : ℂ hz : z ∈ ball z₀ r hgz : g z + ε = 0 e1 : z ≠ z₀ ⊢ (cindex z₀ r fun z => g z + ε) = 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	have e2 : deriv (λ z => g z + ε) z ≠ 0 := by simpa using h14 z (ball_subset_closedBall hz) e1	case h.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ z : ℂ hz : z ∈ ball z₀ r hgz : g z + ε = 0 e1 : z ≠ z₀ ⊢ (cindex z₀ r fun z => g z + ε) = 1	case h.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ z : ℂ hz : z ∈ ball z₀ r hgz : g z + ε = 0 e1 : z ≠ z₀ e2 : deriv (fun z => g z + ε) z ≠ 0 ⊢ (cindex z₀ r fun z => g z + ε) = 1
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	refine crucial hU h20 hz (h18 ε) hgz e2 (λ w hw hwz => ?_)	case h.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ z : ℂ hz : z ∈ ball z₀ r hgz : g z + ε = 0 e1 : z ≠ z₀ e2 : deriv (fun z => g z + ε) z ≠ 0 ⊢ (cindex z₀ r fun z => g z + ε) = 1	case h.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ z : ℂ hz : z ∈ ball z₀ r hgz : g z + ε = 0 e1 : z ≠ z₀ e2 : deriv (fun z => g z + ε) z ≠ 0 w : ℂ hw : w ∈ closedBall z₀ r hwz : w ≠ z ⊢ g w + ε ≠ 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	contrapose! hwz	case h.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ z : ℂ hz : z ∈ ball z₀ r hgz : g z + ε = 0 e1 : z ≠ z₀ e2 : deriv (fun z => g z + ε) z ≠ 0 w : ℂ hw : w ∈ closedBall z₀ r hwz : w ≠ z ⊢ g w + ε ≠ 0	case h.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ z : ℂ hz : z ∈ ball z₀ r hgz : g z + ε = 0 e1 : z ≠ z₀ e2 : deriv (fun z => g z + ε) z ≠ 0 w : ℂ hw : w ∈ closedBall z₀ r hwz : g w + ε = 0 ⊢ w = z
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	exact hi (h20 hw) ((ball_subset_closedBall.trans h20) hz) (add_right_cancel (hwz.trans hgz.symm))	case h.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ z : ℂ hz : z ∈ ball z₀ r hgz : g z + ε = 0 e1 : z ≠ z₀ e2 : deriv (fun z => g z + ε) z ≠ 0 w : ℂ hw : w ∈ closedBall z₀ r hwz : g w + ε = 0 ⊢ w = z	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	linarith	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U ⊢ FormalMultilinearSeries.order p ≠ 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	simp [h8, h24]	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 ⊢ cindex z₀ r g ≠ 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	rintro rfl	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ z : ℂ hz : z ∈ ball z₀ r hgz : g z + ε = 0 ⊢ z ≠ z₀	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U p : FormalMultilinearSeries ℂ ℂ ℂ h17 : p ≠ 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 ε : ℂ hε : ε ∈ {0}ᶜ z : ℂ hgz : g z + ε = 0 hz₀ : z ∈ U hgz₀ : g z = 0 hp : HasFPowerSeriesAt g p z h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z, g z ≠ 0 h✝ : deriv g z = 0 h8 : cindex z r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z_1 ∈ closedBall z r, z_1 ≠ z → deriv g z_1 ≠ 0 h21 : ∀ z_1 ∈ closedBall z r, z_1 ≠ z → g z_1 ≠ 0 h20 : closedBall z r ⊆ U h22 : ∀ z_1 ∈ sphere z r, g z_1 ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z_1 ∈ ball z r, g z_1 + x = 0 h : ε ∈ {0}ᶜ → ∃ z_1 ∈ ball z r, g z_1 + ε = 0 hz : z ∈ ball z r ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	rw [hgz₀, zero_add] at hgz	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U p : FormalMultilinearSeries ℂ ℂ ℂ h17 : p ≠ 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 ε : ℂ hε : ε ∈ {0}ᶜ z : ℂ hgz : g z + ε = 0 hz₀ : z ∈ U hgz₀ : g z = 0 hp : HasFPowerSeriesAt g p z h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z, g z ≠ 0 h✝ : deriv g z = 0 h8 : cindex z r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z_1 ∈ closedBall z r, z_1 ≠ z → deriv g z_1 ≠ 0 h21 : ∀ z_1 ∈ closedBall z r, z_1 ≠ z → g z_1 ≠ 0 h20 : closedBall z r ⊆ U h22 : ∀ z_1 ∈ sphere z r, g z_1 ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z_1 ∈ ball z r, g z_1 + x = 0 h : ε ∈ {0}ᶜ → ∃ z_1 ∈ ball z r, g z_1 + ε = 0 hz : z ∈ ball z r ⊢ False	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U p : FormalMultilinearSeries ℂ ℂ ℂ h17 : p ≠ 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 ε : ℂ hε : ε ∈ {0}ᶜ z : ℂ hgz : ε = 0 hz₀ : z ∈ U hgz₀ : g z = 0 hp : HasFPowerSeriesAt g p z h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z, g z ≠ 0 h✝ : deriv g z = 0 h8 : cindex z r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z_1 ∈ closedBall z r, z_1 ≠ z → deriv g z_1 ≠ 0 h21 : ∀ z_1 ∈ closedBall z r, z_1 ≠ z → g z_1 ≠ 0 h20 : closedBall z r ⊆ U h22 : ∀ z_1 ∈ sphere z r, g z_1 ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z_1 ∈ ball z r, g z_1 + x = 0 h : ε ∈ {0}ᶜ → ∃ z_1 ∈ ball z r, g z_1 + ε = 0 hz : z ∈ ball z r ⊢ False
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	exact hε hgz	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U p : FormalMultilinearSeries ℂ ℂ ℂ h17 : p ≠ 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 ε : ℂ hε : ε ∈ {0}ᶜ z : ℂ hgz : ε = 0 hz₀ : z ∈ U hgz₀ : g z = 0 hp : HasFPowerSeriesAt g p z h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z, g z ≠ 0 h✝ : deriv g z = 0 h8 : cindex z r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z_1 ∈ closedBall z r, z_1 ≠ z → deriv g z_1 ≠ 0 h21 : ∀ z_1 ∈ closedBall z r, z_1 ≠ z → g z_1 ≠ 0 h20 : closedBall z r ⊆ U h22 : ∀ z_1 ∈ sphere z r, g z_1 ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z_1 ∈ ball z r, g z_1 + x = 0 h : ε ∈ {0}ᶜ → ∃ z_1 ∈ ball z r, g z_1 + ε = 0 hz : z ∈ ball z r ⊢ False	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj_aux	[91, 1]	[138, 28]	simpa using h14 z (ball_subset_closedBall hz) e1	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r✝ : ℝ f g✝ g : ℂ → ℂ hU : IsOpen U hg : DifferentiableOn ℂ g U hi : InjOn g U hz₀ : z₀ ∈ U hgz₀ : g z₀ = 0 p : FormalMultilinearSeries ℂ ℂ ℂ hp : HasFPowerSeriesAt g p z₀ h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0 h17 : p ≠ 0 h✝ : deriv g z₀ = 0 h6 : 2 ≤ FormalMultilinearSeries.order p r : ℝ h7 : r > 0 h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p) h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0 h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0 h20 : closedBall z₀ r ⊆ U h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0 h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U h24 : FormalMultilinearSeries.order p ≠ 0 this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0 ε : ℂ h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0 hε : ε ∈ {0}ᶜ z : ℂ hz : z ∈ ball z₀ r hgz : g z + ε = 0 e1 : z ≠ z₀ ⊢ deriv (fun z => g z + ε) z ≠ 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj	[140, 1]	[144, 99]	have : InjOn (λ z => f z - f z₀) U := λ z₁ hz₁ z₂ hz₂ h => hi hz₁ hz₂ (sub_left_inj.1 h)	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f✝ g f : ℂ → ℂ hU : IsOpen U hf : DifferentiableOn ℂ f U hi : InjOn f U hz₀ : z₀ ∈ U ⊢ deriv f z₀ ≠ 0	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f✝ g f : ℂ → ℂ hU : IsOpen U hf : DifferentiableOn ℂ f U hi : InjOn f U hz₀ : z₀ ∈ U this : InjOn (fun z => f z - f z₀) U ⊢ deriv f z₀ ≠ 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/deriv_inj.lean	deriv_ne_zero_of_inj	[140, 1]	[144, 99]	simpa [deriv_sub_const] using deriv_ne_zero_of_inj_aux hU (hf.sub_const _) this hz₀ (sub_self _)	ι : Type u_1 α : Type u_2 β : Type u_3 U : Set ℂ c z₀ : ℂ r : ℝ f✝ g f : ℂ → ℂ hU : IsOpen U hf : DifferentiableOn ℂ f U hi : InjOn f U hz₀ : z₀ ∈ U this : InjOn (fun z => f z - f z₀) U ⊢ deriv f z₀ ≠ 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Spaces.lean	tendsto_𝓒_iff	[15, 1]	[18, 65]	simp [UniformOnFun.tendsto_iff_tendstoUniformlyOn, _root_.compacts]	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U F : ι → 𝓒 U f : 𝓒 U ⊢ Tendsto F l (𝓝 f) ↔ TendstoLocallyUniformlyOn F f l U	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U F : ι → 𝓒 U f : 𝓒 U ⊢ (∀ s ⊆ U, IsCompact s → TendstoUniformlyOn (⇑(UniformOnFun.toFun {K \| K ⊆ U ∧ IsCompact K}) ∘ F) ((UniformOnFun.toFun {K \| K ⊆ U ∧ IsCompact K}) f) l s) ↔ TendstoLocallyUniformlyOn F f l U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Spaces.lean	tendsto_𝓒_iff	[15, 1]	[18, 65]	exact (tendstoLocallyUniformlyOn_iff_forall_isCompact hU).symm	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U F : ι → 𝓒 U f : 𝓒 U ⊢ (∀ s ⊆ U, IsCompact s → TendstoUniformlyOn (⇑(UniformOnFun.toFun {K \| K ⊆ U ∧ IsCompact K}) ∘ F) ((UniformOnFun.toFun {K \| K ⊆ U ∧ IsCompact K}) f) l s) ↔ TendstoLocallyUniformlyOn F f l U	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Spaces.lean	isClosed_𝓗	[24, 1]	[28, 79]	refine isClosed_iff_clusterPt.2 (λ f hf => ?_)	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U ⊢ IsClosed (𝓗 U)	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U hf : ClusterPt f (𝓟 (𝓗 U)) ⊢ f ∈ 𝓗 U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Spaces.lean	isClosed_𝓗	[24, 1]	[28, 79]	refine @TendstoLocallyUniformlyOn.differentiableOn _ _ _ _ _ _ _ id f hf ?_ ?_ hU	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U hf : ClusterPt f (𝓟 (𝓗 U)) ⊢ f ∈ 𝓗 U	case refine_1 U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U hf : ClusterPt f (𝓟 (𝓗 U)) ⊢ TendstoLocallyUniformlyOn id f (𝓝 f ⊓ 𝓟 (𝓗 U)) U case refine_2 U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U hf : ClusterPt f (𝓟 (𝓗 U)) ⊢ ∀ᶠ (n : ℂ → ℂ) in 𝓝 f ⊓ 𝓟 (𝓗 U), DifferentiableOn ℂ (id n) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Spaces.lean	isClosed_𝓗	[24, 1]	[28, 79]	simp [← tendsto_𝓒_iff hU, Tendsto]	case refine_1 U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U hf : ClusterPt f (𝓟 (𝓗 U)) ⊢ TendstoLocallyUniformlyOn id f (𝓝 f ⊓ 𝓟 (𝓗 U)) U	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Spaces.lean	isClosed_𝓗	[24, 1]	[28, 79]	simp [eventually_inf_principal, 𝓗]	case refine_2 U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U hf : ClusterPt f (𝓟 (𝓗 U)) ⊢ ∀ᶠ (n : ℂ → ℂ) in 𝓝 f ⊓ 𝓟 (𝓗 U), DifferentiableOn ℂ (id n) U	case refine_2 U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U hf : ClusterPt f (𝓟 (𝓗 U)) ⊢ ∀ᶠ (x : ℂ → ℂ) in 𝓝 f, x ∈ {f \| DifferentiableOn ℂ f U} → DifferentiableOn ℂ x U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Spaces.lean	isClosed_𝓗	[24, 1]	[28, 79]	exact eventually_of_forall (λ g => id)	case refine_2 U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U hf : ClusterPt f (𝓟 (𝓗 U)) ⊢ ∀ᶠ (x : ℂ → ℂ) in 𝓝 f, x ∈ {f \| DifferentiableOn ℂ f U} → DifferentiableOn ℂ x U	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Spaces.lean	ContinuousOn_uderiv	[30, 1]	[35, 27]	rintro f -	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U ⊢ ContinuousOn uderiv (𝓗 U)	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U ⊢ ContinuousWithinAt uderiv (𝓗 U) f
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Spaces.lean	ContinuousOn_uderiv	[30, 1]	[35, 27]	refine (tendsto_𝓒_iff hU).2 ?_	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U ⊢ ContinuousWithinAt uderiv (𝓗 U) f	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U ⊢ TendstoLocallyUniformlyOn uderiv (uderiv f) (𝓝[𝓗 U] f) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Spaces.lean	ContinuousOn_uderiv	[30, 1]	[35, 27]	refine TendstoLocallyUniformlyOn.deriv ?_ eventually_mem_nhdsWithin hU	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U ⊢ TendstoLocallyUniformlyOn uderiv (uderiv f) (𝓝[𝓗 U] f) U	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U ⊢ TendstoLocallyUniformlyOn (fun f => f) f (𝓝[𝓗 U] f) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/Spaces.lean	ContinuousOn_uderiv	[30, 1]	[35, 27]	apply (tendsto_𝓒_iff hU).1	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U ⊢ TendstoLocallyUniformlyOn (fun f => f) f (𝓝[𝓗 U] f) U	U : Set ℂ Q : Set ℂ → Set ℂ ι : Type u_1 l : Filter ι hU : IsOpen U f : 𝓒 U ⊢ Tendsto (fun f => f) (𝓝[𝓗 U] f) (𝓝 f)

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