Datasets:
pkuAI4M
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lean_github

Dataset card Data Studio Files Community
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https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.inv_eq_inf
[102, 9]
[105, 89]
simp only [inv_inf]
case h₁ ⊢ ∞⁻¹ = ∞ ↔ ∞ = 0
case h₁ ⊢ 0 = ∞ ↔ ∞ = 0
Please generate a tactic in lean4 to solve the state. STATE: case h₁ ⊢ ∞⁻¹ = ∞ ↔ ∞ = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.inv_eq_inf
[102, 9]
[105, 89]
exact ⟨Eq.symm, Eq.symm⟩
case h₁ ⊢ 0 = ∞ ↔ ∞ = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h₁ ⊢ 0 = ∞ ↔ ∞ = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.inv_eq_inf
[102, 9]
[105, 89]
simp only [inv_def, inv, not_not, imp_false, ite_eq_left_iff, OnePoint.coe_ne_infty]
case h₂ x✝ : ℂ ⊢ (↑x✝)⁻¹ = ∞ ↔ ↑x✝ = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h₂ x✝ : ℂ ⊢ (↑x✝)⁻¹ = ∞ ↔ ↑x✝ = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.inv_eq_zero
[106, 9]
[112, 38]
induction' z using OnePoint.rec with z
z : 𝕊 ⊢ z⁻¹ = 0 ↔ z = ∞
case h₁ ⊢ ∞⁻¹ = 0 ↔ ∞ = ∞ case h₂ z : ℂ ⊢ (↑z)⁻¹ = 0 ↔ ↑z = ∞
Please generate a tactic in lean4 to solve the state. STATE: z : 𝕊 ⊢ z⁻¹ = 0 ↔ z = ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.inv_eq_zero
[106, 9]
[112, 38]
simp only [inv_inf, eq_self_iff_true]
case h₁ ⊢ ∞⁻¹ = 0 ↔ ∞ = ∞
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h₁ ⊢ ∞⁻¹ = 0 ↔ ∞ = ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.inv_eq_zero
[106, 9]
[112, 38]
simp only [inv_def, inv, toComplex_coe]
case h₂ z : ℂ ⊢ (↑z)⁻¹ = 0 ↔ ↑z = ∞
case h₂ z : ℂ ⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞
Please generate a tactic in lean4 to solve the state. STATE: case h₂ z : ℂ ⊢ (↑z)⁻¹ = 0 ↔ ↑z = ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.inv_eq_zero
[106, 9]
[112, 38]
by_cases z0 : (z : 𝕊) = 0
case h₂ z : ℂ ⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞
case pos z : ℂ z0 : ↑z = 0 ⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞ case neg z : ℂ z0 : ¬↑z = 0 ⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞
Please generate a tactic in lean4 to solve the state. STATE: case h₂ z : ℂ ⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.inv_eq_zero
[106, 9]
[112, 38]
simp only [if_pos, z0, inf_ne_zero, inf_ne_zero.symm]
case pos z : ℂ z0 : ↑z = 0 ⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞ case neg z : ℂ z0 : ¬↑z = 0 ⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞
case neg z : ℂ z0 : ¬↑z = 0 ⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞
Please generate a tactic in lean4 to solve the state. STATE: case pos z : ℂ z0 : ↑z = 0 ⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞ case neg z : ℂ z0 : ¬↑z = 0 ⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.inv_eq_zero
[106, 9]
[112, 38]
simp only [if_neg z0, coe_ne_inf, iff_false_iff]
case neg z : ℂ z0 : ¬↑z = 0 ⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞
case neg z : ℂ z0 : ¬↑z = 0 ⊢ ¬↑z⁻¹ = 0
Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℂ z0 : ¬↑z = 0 ⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.inv_eq_zero
[106, 9]
[112, 38]
rw [coe_eq_zero, _root_.inv_eq_zero]
case neg z : ℂ z0 : ¬↑z = 0 ⊢ ¬↑z⁻¹ = 0
case neg z : ℂ z0 : ¬↑z = 0 ⊢ ¬z = 0
Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℂ z0 : ¬↑z = 0 ⊢ ¬↑z⁻¹ = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.inv_eq_zero
[106, 9]
[112, 38]
simpa only [coe_eq_zero] using z0
case neg z : ℂ z0 : ¬↑z = 0 ⊢ ¬z = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℂ z0 : ¬↑z = 0 ⊢ ¬z = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.toComplex_inv
[113, 1]
[118, 64]
induction' z using OnePoint.rec with z
z : 𝕊 ⊢ z⁻¹.toComplex = z.toComplex⁻¹
case h₁ ⊢ ∞⁻¹.toComplex = ∞.toComplex⁻¹ case h₂ z : ℂ ⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹
Please generate a tactic in lean4 to solve the state. STATE: z : 𝕊 ⊢ z⁻¹.toComplex = z.toComplex⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.toComplex_inv
[113, 1]
[118, 64]
simp only [inv_inf, toComplex_zero, toComplex_inf, inv_zero', inv_zero, eq_self_iff_true]
case h₁ ⊢ ∞⁻¹.toComplex = ∞.toComplex⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h₁ ⊢ ∞⁻¹.toComplex = ∞.toComplex⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.toComplex_inv
[113, 1]
[118, 64]
by_cases z0 : z = 0
case h₂ z : ℂ ⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹
case pos z : ℂ z0 : z = 0 ⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹ case neg z : ℂ z0 : ¬z = 0 ⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case h₂ z : ℂ ⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.toComplex_inv
[113, 1]
[118, 64]
simp only [z0, coe_zero, inv_zero', toComplex_inf, toComplex_zero, inv_zero]
case pos z : ℂ z0 : z = 0 ⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos z : ℂ z0 : z = 0 ⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.toComplex_inv
[113, 1]
[118, 64]
simp only [z0, inv_coe, Ne, not_false_iff, toComplex_coe]
case neg z : ℂ z0 : ¬z = 0 ⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℂ z0 : ¬z = 0 ⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.coe_tendsto_inf
[121, 1]
[123, 27]
rw [Filter.tendsto_iff_comap, OnePoint.comap_coe_nhds_infty, Filter.coclosedCompact_eq_cocompact]
⊢ Tendsto (fun z => ↑z) atInf (𝓝 ∞)
⊢ atInf ≤ Filter.cocompact ℂ
Please generate a tactic in lean4 to solve the state. STATE: ⊢ Tendsto (fun z => ↑z) atInf (𝓝 ∞) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.coe_tendsto_inf
[121, 1]
[123, 27]
exact atInf_le_cocompact
⊢ atInf ≤ Filter.cocompact ℂ
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊢ atInf ≤ Filter.cocompact ℂ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.coe_tendsto_inf'
[126, 1]
[133, 59]
simp only [e, tendsto_nhdsWithin_range, coe_tendsto_inf]
e : {∞}ᶜ = range fun z => ↑z ⊢ Tendsto (fun z => ↑z) atInf (𝓝[≠] ∞)
no goals
Please generate a tactic in lean4 to solve the state. STATE: e : {∞}ᶜ = range fun z => ↑z ⊢ Tendsto (fun z => ↑z) atInf (𝓝[≠] ∞) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.coe_tendsto_inf'
[126, 1]
[133, 59]
ext z
⊢ {∞}ᶜ = range fun z => ↑z
case h z : 𝕊 ⊢ z ∈ {∞}ᶜ ↔ z ∈ range fun z => ↑z
Please generate a tactic in lean4 to solve the state. STATE: ⊢ {∞}ᶜ = range fun z => ↑z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.coe_tendsto_inf'
[126, 1]
[133, 59]
induction' z using OnePoint.rec with z
case h z : 𝕊 ⊢ z ∈ {∞}ᶜ ↔ z ∈ range fun z => ↑z
case h.h₁ ⊢ ∞ ∈ {∞}ᶜ ↔ ∞ ∈ range fun z => ↑z case h.h₂ z : ℂ ⊢ ↑z ∈ {∞}ᶜ ↔ ↑z ∈ range fun z => ↑z
Please generate a tactic in lean4 to solve the state. STATE: case h z : 𝕊 ⊢ z ∈ {∞}ᶜ ↔ z ∈ range fun z => ↑z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.coe_tendsto_inf'
[126, 1]
[133, 59]
simp only [mem_compl_iff, mem_singleton_iff, not_true, mem_range, OnePoint.coe_ne_infty, exists_false]
case h.h₁ ⊢ ∞ ∈ {∞}ᶜ ↔ ∞ ∈ range fun z => ↑z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h₁ ⊢ ∞ ∈ {∞}ᶜ ↔ ∞ ∈ range fun z => ↑z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.coe_tendsto_inf'
[126, 1]
[133, 59]
simp only [mem_compl_iff, mem_singleton_iff, OnePoint.coe_ne_infty, not_false_eq_true, mem_range, coe_eq_coe, exists_eq]
case h.h₂ z : ℂ ⊢ ↑z ∈ {∞}ᶜ ↔ ↑z ∈ range fun z => ↑z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h₂ z : ℂ ⊢ ↑z ∈ {∞}ᶜ ↔ ↑z ∈ range fun z => ↑z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
rw [continuous_iff_continuousOn_univ]
⊢ Continuous fun z => z⁻¹
⊢ ContinuousOn (fun z => z⁻¹) univ
Please generate a tactic in lean4 to solve the state. STATE: ⊢ Continuous fun z => z⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
intro z _
⊢ ContinuousOn (fun z => z⁻¹) univ
z : 𝕊 a✝ : z ∈ univ ⊢ ContinuousWithinAt (fun z => z⁻¹) univ z
Please generate a tactic in lean4 to solve the state. STATE: ⊢ ContinuousOn (fun z => z⁻¹) univ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
apply ContinuousAt.continuousWithinAt
z : 𝕊 a✝ : z ∈ univ ⊢ ContinuousWithinAt (fun z => z⁻¹) univ z
case h z : 𝕊 a✝ : z ∈ univ ⊢ ContinuousAt (fun z => z⁻¹) z
Please generate a tactic in lean4 to solve the state. STATE: z : 𝕊 a✝ : z ∈ univ ⊢ ContinuousWithinAt (fun z => z⁻¹) univ z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
induction' z using OnePoint.rec with z
case h z : 𝕊 a✝ : z ∈ univ ⊢ ContinuousAt (fun z => z⁻¹) z
case h.h₁ a✝ : ∞ ∈ univ ⊢ ContinuousAt (fun z => z⁻¹) ∞ case h.h₂ z : ℂ a✝ : ↑z ∈ univ ⊢ ContinuousAt (fun z => z⁻¹) ↑z
Please generate a tactic in lean4 to solve the state. STATE: case h z : 𝕊 a✝ : z ∈ univ ⊢ ContinuousAt (fun z => z⁻¹) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
simp only [OnePoint.continuousAt_infty', Function.comp, Filter.coclosedCompact_eq_cocompact, inv_inf, ← atInf_eq_cocompact]
case h.h₁ a✝ : ∞ ∈ univ ⊢ ContinuousAt (fun z => z⁻¹) ∞
case h.h₁ a✝ : ∞ ∈ univ ⊢ Tendsto (fun x => (↑x)⁻¹) atInf (𝓝 0)
Please generate a tactic in lean4 to solve the state. STATE: case h.h₁ a✝ : ∞ ∈ univ ⊢ ContinuousAt (fun z => z⁻¹) ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
have e : ∀ᶠ z : ℂ in atInf, ↑z⁻¹ = (↑z : 𝕊)⁻¹ := by refine (eventually_atInf 0).mp (eventually_of_forall fun z z0 ↦ ?_) simp only [gt_iff_lt, Complex.norm_eq_abs, AbsoluteValue.pos_iff] at z0; rw [inv_coe z0]
case h.h₁ a✝ : ∞ ∈ univ ⊢ Tendsto (fun x => (↑x)⁻¹) atInf (𝓝 0)
case h.h₁ a✝ : ∞ ∈ univ e : ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹ ⊢ Tendsto (fun x => (↑x)⁻¹) atInf (𝓝 0)
Please generate a tactic in lean4 to solve the state. STATE: case h.h₁ a✝ : ∞ ∈ univ ⊢ Tendsto (fun x => (↑x)⁻¹) atInf (𝓝 0) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
apply Filter.Tendsto.congr' e
case h.h₁ a✝ : ∞ ∈ univ e : ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹ ⊢ Tendsto (fun x => (↑x)⁻¹) atInf (𝓝 0)
case h.h₁ a✝ : ∞ ∈ univ e : ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹ ⊢ Tendsto (fun x => ↑x⁻¹) atInf (𝓝 0)
Please generate a tactic in lean4 to solve the state. STATE: case h.h₁ a✝ : ∞ ∈ univ e : ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹ ⊢ Tendsto (fun x => (↑x)⁻¹) atInf (𝓝 0) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
exact Filter.Tendsto.comp continuous_coe.continuousAt inv_tendsto_atInf'
case h.h₁ a✝ : ∞ ∈ univ e : ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹ ⊢ Tendsto (fun x => ↑x⁻¹) atInf (𝓝 0)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h₁ a✝ : ∞ ∈ univ e : ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹ ⊢ Tendsto (fun x => ↑x⁻¹) atInf (𝓝 0) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
refine (eventually_atInf 0).mp (eventually_of_forall fun z z0 ↦ ?_)
a✝ : ∞ ∈ univ ⊢ ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹
a✝ : ∞ ∈ univ z : ℂ z0 : ‖z‖ > 0 ⊢ ↑z⁻¹ = (↑z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: a✝ : ∞ ∈ univ ⊢ ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
simp only [gt_iff_lt, Complex.norm_eq_abs, AbsoluteValue.pos_iff] at z0
a✝ : ∞ ∈ univ z : ℂ z0 : ‖z‖ > 0 ⊢ ↑z⁻¹ = (↑z)⁻¹
a✝ : ∞ ∈ univ z : ℂ z0 : z ≠ 0 ⊢ ↑z⁻¹ = (↑z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: a✝ : ∞ ∈ univ z : ℂ z0 : ‖z‖ > 0 ⊢ ↑z⁻¹ = (↑z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
rw [inv_coe z0]
a✝ : ∞ ∈ univ z : ℂ z0 : z ≠ 0 ⊢ ↑z⁻¹ = (↑z)⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: a✝ : ∞ ∈ univ z : ℂ z0 : z ≠ 0 ⊢ ↑z⁻¹ = (↑z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
simp only [OnePoint.continuousAt_coe, Function.comp, inv_def, inv, WithTop.coe_eq_zero, toComplex_coe]
case h.h₂ z : ℂ a✝ : ↑z ∈ univ ⊢ ContinuousAt (fun z => z⁻¹) ↑z
case h.h₂ z : ℂ a✝ : ↑z ∈ univ ⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z
Please generate a tactic in lean4 to solve the state. STATE: case h.h₂ z : ℂ a✝ : ↑z ∈ univ ⊢ ContinuousAt (fun z => z⁻¹) ↑z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
by_cases z0 : z = 0
case h.h₂ z : ℂ a✝ : ↑z ∈ univ ⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z
case pos z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z case neg z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 ⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z
Please generate a tactic in lean4 to solve the state. STATE: case h.h₂ z : ℂ a✝ : ↑z ∈ univ ⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
simp only [z0, ContinuousAt, OnePoint.nhds_infty_eq, eq_self_iff_true, if_true, Filter.coclosedCompact_eq_cocompact]
case pos z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z
case pos z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝 0) (𝓝 (if ↑0 = 0 then ∞ else ↑0⁻¹))
Please generate a tactic in lean4 to solve the state. STATE: case pos z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
simp only [← nhdsWithin_compl_singleton_sup_pure, Filter.tendsto_sup]
case pos z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝 0) (𝓝 (if ↑0 = 0 then ∞ else ↑0⁻¹))
case pos z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0) ((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) ∧ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0) ((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹))
Please generate a tactic in lean4 to solve the state. STATE: case pos z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝 0) (𝓝 (if ↑0 = 0 then ∞ else ↑0⁻¹)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
constructor
case pos z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0) ((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) ∧ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0) ((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹))
case pos.left z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0) ((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) case pos.right z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0) ((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹))
Please generate a tactic in lean4 to solve the state. STATE: case pos z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0) ((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) ∧ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0) ((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
refine Filter.Tendsto.mono_right ?_ le_sup_left
case pos.left z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0) ((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹))
case pos.left z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0) (𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹)
Please generate a tactic in lean4 to solve the state. STATE: case pos.left z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0) ((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
apply tendsto_nhdsWithin_congr (f := fun z : ℂ ↦ (↑z⁻¹ : 𝕊))
case pos.left z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0) (𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹)
case pos.left.hfg z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ ∀ x ∈ {0}ᶜ, ↑x⁻¹ = if ↑x = 0 then ∞ else ↑x⁻¹ case pos.left.hf z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun z => ↑z⁻¹) (𝓝[≠] 0) (𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹)
Please generate a tactic in lean4 to solve the state. STATE: case pos.left z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0) (𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
intro z m
case pos.left.hfg z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ ∀ x ∈ {0}ᶜ, ↑x⁻¹ = if ↑x = 0 then ∞ else ↑x⁻¹
case pos.left.hfg z✝ : ℂ a✝ : ↑z✝ ∈ univ z0 : z✝ = 0 z : ℂ m : z ∈ {0}ᶜ ⊢ ↑z⁻¹ = if ↑z = 0 then ∞ else ↑z⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case pos.left.hfg z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ ∀ x ∈ {0}ᶜ, ↑x⁻¹ = if ↑x = 0 then ∞ else ↑x⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
rw [mem_compl_singleton_iff] at m
case pos.left.hfg z✝ : ℂ a✝ : ↑z✝ ∈ univ z0 : z✝ = 0 z : ℂ m : z ∈ {0}ᶜ ⊢ ↑z⁻¹ = if ↑z = 0 then ∞ else ↑z⁻¹
case pos.left.hfg z✝ : ℂ a✝ : ↑z✝ ∈ univ z0 : z✝ = 0 z : ℂ m : z ≠ 0 ⊢ ↑z⁻¹ = if ↑z = 0 then ∞ else ↑z⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case pos.left.hfg z✝ : ℂ a✝ : ↑z✝ ∈ univ z0 : z✝ = 0 z : ℂ m : z ∈ {0}ᶜ ⊢ ↑z⁻¹ = if ↑z = 0 then ∞ else ↑z⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
simp only [coe_eq_zero, m, ite_false]
case pos.left.hfg z✝ : ℂ a✝ : ↑z✝ ∈ univ z0 : z✝ = 0 z : ℂ m : z ≠ 0 ⊢ ↑z⁻¹ = if ↑z = 0 then ∞ else ↑z⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos.left.hfg z✝ : ℂ a✝ : ↑z✝ ∈ univ z0 : z✝ = 0 z : ℂ m : z ≠ 0 ⊢ ↑z⁻¹ = if ↑z = 0 then ∞ else ↑z⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
simp only [coe_zero, ite_true]
case pos.left.hf z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun z => ↑z⁻¹) (𝓝[≠] 0) (𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹)
case pos.left.hf z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun z => ↑z⁻¹) (𝓝[≠] 0) (𝓝[≠] ∞)
Please generate a tactic in lean4 to solve the state. STATE: case pos.left.hf z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun z => ↑z⁻¹) (𝓝[≠] 0) (𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
apply coe_tendsto_inf'.comp
case pos.left.hf z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun z => ↑z⁻¹) (𝓝[≠] 0) (𝓝[≠] ∞)
case pos.left.hf z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun z => z⁻¹) (𝓝[≠] 0) atInf
Please generate a tactic in lean4 to solve the state. STATE: case pos.left.hf z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun z => ↑z⁻¹) (𝓝[≠] 0) (𝓝[≠] ∞) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
rw [← @tendsto_atInf_iff_tendsto_nhds_zero ℂ ℂ _ _ fun z : ℂ ↦ z]
case pos.left.hf z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun z => z⁻¹) (𝓝[≠] 0) atInf
case pos.left.hf z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun z => z) atInf atInf
Please generate a tactic in lean4 to solve the state. STATE: case pos.left.hf z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun z => z⁻¹) (𝓝[≠] 0) atInf TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
exact Filter.tendsto_id
case pos.left.hf z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun z => z) atInf atInf
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos.left.hf z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun z => z) atInf atInf TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
refine Filter.Tendsto.mono_right ?_ le_sup_right
case pos.right z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0) ((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹))
case pos.right z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0) (pure (if ↑0 = 0 then ∞ else ↑0⁻¹))
Please generate a tactic in lean4 to solve the state. STATE: case pos.right z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0) ((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
simp only [Filter.pure_zero, Filter.tendsto_pure, ite_eq_left_iff, Filter.eventually_zero, eq_self_iff_true, not_true, IsEmpty.forall_iff]
case pos.right z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0) (pure (if ↑0 = 0 then ∞ else ↑0⁻¹))
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos.right z : ℂ a✝ : ↑z ∈ univ z0 : z = 0 ⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0) (pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
have e : ∀ᶠ w : ℂ in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹ : 𝕊) = ↑w⁻¹ := by refine (continuousAt_id.eventually_ne z0).mp (eventually_of_forall fun w w0 ↦ ?_) simp only [Ne, id_eq] at w0; simp only [w0, if_false]
case neg z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 ⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z
case neg z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 e : ∀ᶠ (w : ℂ) in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹ ⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z
Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 ⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
simp only [coe_eq_zero, continuousAt_congr e]
case neg z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 e : ∀ᶠ (w : ℂ) in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹ ⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z
case neg z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 e : ∀ᶠ (w : ℂ) in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹ ⊢ ContinuousAt (fun x => ↑x⁻¹) z
Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 e : ∀ᶠ (w : ℂ) in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹ ⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
exact continuous_coe.continuousAt.comp (tendsto_inv₀ z0)
case neg z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 e : ∀ᶠ (w : ℂ) in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹ ⊢ ContinuousAt (fun x => ↑x⁻¹) z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 e : ∀ᶠ (w : ℂ) in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹ ⊢ ContinuousAt (fun x => ↑x⁻¹) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
refine (continuousAt_id.eventually_ne z0).mp (eventually_of_forall fun w w0 ↦ ?_)
z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 ⊢ ∀ᶠ (w : ℂ) in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹
z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 w : ℂ w0 : id w ≠ 0 ⊢ (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 ⊢ ∀ᶠ (w : ℂ) in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
simp only [Ne, id_eq] at w0
z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 w : ℂ w0 : id w ≠ 0 ⊢ (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹
z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 w : ℂ w0 : ¬w = 0 ⊢ (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 w : ℂ w0 : id w ≠ 0 ⊢ (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.continuous_inv
[136, 1]
[166, 63]
simp only [w0, if_false]
z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 w : ℂ w0 : ¬w = 0 ⊢ (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ a✝ : ↑z ∈ univ z0 : ¬z = 0 w : ℂ w0 : ¬w = 0 ⊢ (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.invEquiv_apply
[181, 9]
[182, 40]
simp only [invEquiv, Equiv.coe_fn_mk]
z : 𝕊 ⊢ invEquiv z = z⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : 𝕊 ⊢ invEquiv z = z⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.invEquiv_symm
[183, 9]
[185, 18]
simp only [Equiv.ext_iff, invEquiv, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, eq_self_iff_true, forall_const]
⊢ invEquiv.symm = invEquiv
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊢ invEquiv.symm = invEquiv TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.invHomeomorph_apply
[186, 9]
[187, 74]
simp only [invHomeomorph, Homeomorph.homeomorph_mk_coe, invEquiv_apply]
z : 𝕊 ⊢ invHomeomorph z = z⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : 𝕊 ⊢ invHomeomorph z = z⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.invHomeomorph_symm
[188, 9]
[190, 67]
simp only [invHomeomorph, Homeomorph.homeomorph_mk_coe_symm, invEquiv_symm, Homeomorph.homeomorph_mk_coe, eq_self_iff_true, forall_const]
⊢ ∀ (x : 𝕊), invHomeomorph.symm x = invHomeomorph x
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (x : 𝕊), invHomeomorph.symm x = invHomeomorph x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.coePartialHomeomorph_target
[217, 1]
[218, 59]
simp only [coePartialHomeomorph, coePartialEquiv_target]
⊢ coePartialHomeomorph.target = {∞}ᶜ
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊢ coePartialHomeomorph.target = {∞}ᶜ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.invCoePartialHomeomorph_target
[219, 1]
[224, 16]
ext z
⊢ invCoePartialHomeomorph.target = {0}ᶜ
case h z : 𝕊 ⊢ z ∈ invCoePartialHomeomorph.target ↔ z ∈ {0}ᶜ
Please generate a tactic in lean4 to solve the state. STATE: ⊢ invCoePartialHomeomorph.target = {0}ᶜ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.invCoePartialHomeomorph_target
[219, 1]
[224, 16]
simp only [invCoePartialHomeomorph, PartialHomeomorph.trans_toPartialEquiv, PartialEquiv.trans_target, Homeomorph.toPartialHomeomorph_target, PartialHomeomorph.coe_coe_symm, Homeomorph.toPartialHomeomorph_symm_apply, invHomeomorph_symm, coePartialHomeomorph_target, preimage_compl, univ_inter, mem_compl_iff, mem_preimage, invHomeomorph_apply, mem_singleton_iff, inv_eq_inf]
case h z : 𝕊 ⊢ z ∈ invCoePartialHomeomorph.target ↔ z ∈ {0}ᶜ
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h z : 𝕊 ⊢ z ∈ invCoePartialHomeomorph.target ↔ z ∈ {0}ᶜ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.extChartAt_coe
[255, 1]
[258, 29]
simp only [coePartialHomeomorph, extChartAt, PartialHomeomorph.extend, chartAt_coe, PartialHomeomorph.symm_toPartialEquiv, modelWithCornersSelf_partialEquiv, PartialEquiv.trans_refl]
z : ℂ ⊢ extChartAt I ↑z = coePartialEquiv.symm
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ ⊢ extChartAt I ↑z = coePartialEquiv.symm TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.extChartAt_zero
[259, 1]
[260, 41]
simp only [← coe_zero, extChartAt_coe]
⊢ extChartAt I 0 = coePartialEquiv.symm
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊢ extChartAt I 0 = coePartialEquiv.symm TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.extChartAt_inf
[261, 1]
[285, 63]
apply PartialEquiv.ext
⊢ extChartAt I ∞ = invEquiv.toPartialEquiv.trans coePartialEquiv.symm
case h ⊢ ∀ (x : 𝕊), ↑(extChartAt I ∞) x = ↑(invEquiv.toPartialEquiv.trans coePartialEquiv.symm) x case hsymm ⊢ ∀ (x : ℂ), ↑(extChartAt I ∞).symm x = ↑(invEquiv.toPartialEquiv.trans coePartialEquiv.symm).symm x case hs ⊢ (extChartAt I ∞).source = (invEquiv.toPartialEquiv.trans coePartialEquiv.symm).source
Please generate a tactic in lean4 to solve the state. STATE: ⊢ extChartAt I ∞ = invEquiv.toPartialEquiv.trans coePartialEquiv.symm TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.extChartAt_inf
[261, 1]
[285, 63]
intro z
case h ⊢ ∀ (x : 𝕊), ↑(extChartAt I ∞) x = ↑(invEquiv.toPartialEquiv.trans coePartialEquiv.symm) x
case h z : 𝕊 ⊢ ↑(extChartAt I ∞) z = ↑(invEquiv.toPartialEquiv.trans coePartialEquiv.symm) z
Please generate a tactic in lean4 to solve the state. STATE: case h ⊢ ∀ (x : 𝕊), ↑(extChartAt I ∞) x = ↑(invEquiv.toPartialEquiv.trans coePartialEquiv.symm) x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.extChartAt_inf
[261, 1]
[285, 63]
simp only [extChartAt, invCoePartialHomeomorph, coePartialHomeomorph, invHomeomorph, PartialHomeomorph.extend, chartAt_inf, PartialHomeomorph.symm_toPartialEquiv, PartialHomeomorph.trans_toPartialEquiv, modelWithCornersSelf_partialEquiv, PartialEquiv.trans_refl, PartialEquiv.coe_trans_symm, PartialHomeomorph.coe_coe_symm, Homeomorph.toPartialHomeomorph_symm_apply, Homeomorph.homeomorph_mk_coe_symm, invEquiv_symm, PartialEquiv.coe_trans, Equiv.toPartialEquiv_apply]
case h z : 𝕊 ⊢ ↑(extChartAt I ∞) z = ↑(invEquiv.toPartialEquiv.trans coePartialEquiv.symm) z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h z : 𝕊 ⊢ ↑(extChartAt I ∞) z = ↑(invEquiv.toPartialEquiv.trans coePartialEquiv.symm) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.extChartAt_inf
[261, 1]
[285, 63]
intro z
case hsymm ⊢ ∀ (x : ℂ), ↑(extChartAt I ∞).symm x = ↑(invEquiv.toPartialEquiv.trans coePartialEquiv.symm).symm x
case hsymm z : ℂ ⊢ ↑(extChartAt I ∞).symm z = ↑(invEquiv.toPartialEquiv.trans coePartialEquiv.symm).symm z
Please generate a tactic in lean4 to solve the state. STATE: case hsymm ⊢ ∀ (x : ℂ), ↑(extChartAt I ∞).symm x = ↑(invEquiv.toPartialEquiv.trans coePartialEquiv.symm).symm x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.extChartAt_inf
[261, 1]
[285, 63]
simp only [extChartAt, invCoePartialHomeomorph, coePartialHomeomorph, invHomeomorph, invEquiv, PartialHomeomorph.extend, chartAt_inf, PartialHomeomorph.symm_toPartialEquiv, PartialHomeomorph.trans_toPartialEquiv, modelWithCornersSelf_partialEquiv, PartialEquiv.trans_refl, PartialEquiv.symm_symm, PartialEquiv.coe_trans, PartialHomeomorph.coe_coe, Homeomorph.toPartialHomeomorph_apply, Homeomorph.homeomorph_mk_coe, Equiv.coe_fn_mk, PartialEquiv.coe_trans_symm, Equiv.toPartialEquiv_symm_apply, Equiv.coe_fn_symm_mk]
case hsymm z : ℂ ⊢ ↑(extChartAt I ∞).symm z = ↑(invEquiv.toPartialEquiv.trans coePartialEquiv.symm).symm z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hsymm z : ℂ ⊢ ↑(extChartAt I ∞).symm z = ↑(invEquiv.toPartialEquiv.trans coePartialEquiv.symm).symm z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.extChartAt_inf
[261, 1]
[285, 63]
simp only [extChartAt, invCoePartialHomeomorph, coePartialHomeomorph, invHomeomorph, PartialHomeomorph.extend, chartAt_inf, PartialHomeomorph.symm_toPartialEquiv, PartialHomeomorph.trans_toPartialEquiv, modelWithCornersSelf_partialEquiv, PartialEquiv.trans_refl, PartialEquiv.symm_source, PartialEquiv.trans_target, Homeomorph.toPartialHomeomorph_target, PartialHomeomorph.coe_coe_symm, Homeomorph.toPartialHomeomorph_symm_apply, Homeomorph.homeomorph_mk_coe_symm, invEquiv_symm, PartialEquiv.trans_source, Equiv.toPartialEquiv_source, Equiv.toPartialEquiv_apply]
case hs ⊢ (extChartAt I ∞).source = (invEquiv.toPartialEquiv.trans coePartialEquiv.symm).source
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hs ⊢ (extChartAt I ∞).source = (invEquiv.toPartialEquiv.trans coePartialEquiv.symm).source TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.extChartAt_inf_apply
[286, 1]
[288, 48]
simp only [extChartAt_inf, PartialEquiv.trans_apply, coePartialEquiv_symm_apply, Equiv.toPartialEquiv_apply, invEquiv_apply]
x : 𝕊 ⊢ ↑(extChartAt I ∞) x = x⁻¹.toComplex
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : 𝕊 ⊢ ↑(extChartAt I ∞) x = x⁻¹.toComplex TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.tendsto_inf_iff_tendsto_atInf
[318, 1]
[325, 41]
constructor
X : Type f : Filter X g : X → ℂ ⊢ Tendsto (fun x => ↑(g x)) f (𝓝 ∞) ↔ Tendsto (fun x => g x) f atInf
case mp X : Type f : Filter X g : X → ℂ ⊢ Tendsto (fun x => ↑(g x)) f (𝓝 ∞) → Tendsto (fun x => g x) f atInf case mpr X : Type f : Filter X g : X → ℂ ⊢ Tendsto (fun x => g x) f atInf → Tendsto (fun x => ↑(g x)) f (𝓝 ∞)
Please generate a tactic in lean4 to solve the state. STATE: X : Type f : Filter X g : X → ℂ ⊢ Tendsto (fun x => ↑(g x)) f (𝓝 ∞) ↔ Tendsto (fun x => g x) f atInf TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.tendsto_inf_iff_tendsto_atInf
[318, 1]
[325, 41]
intro t
case mp X : Type f : Filter X g : X → ℂ ⊢ Tendsto (fun x => ↑(g x)) f (𝓝 ∞) → Tendsto (fun x => g x) f atInf
case mp X : Type f : Filter X g : X → ℂ t : Tendsto (fun x => ↑(g x)) f (𝓝 ∞) ⊢ Tendsto (fun x => g x) f atInf
Please generate a tactic in lean4 to solve the state. STATE: case mp X : Type f : Filter X g : X → ℂ ⊢ Tendsto (fun x => ↑(g x)) f (𝓝 ∞) → Tendsto (fun x => g x) f atInf TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.tendsto_inf_iff_tendsto_atInf
[318, 1]
[325, 41]
simp only [Filter.tendsto_iff_comap] at t ⊢
case mp X : Type f : Filter X g : X → ℂ t : Tendsto (fun x => ↑(g x)) f (𝓝 ∞) ⊢ Tendsto (fun x => g x) f atInf
case mp X : Type f : Filter X g : X → ℂ t : f ≤ Filter.comap (fun x => ↑(g x)) (𝓝 ∞) ⊢ f ≤ Filter.comap (fun x => g x) atInf
Please generate a tactic in lean4 to solve the state. STATE: case mp X : Type f : Filter X g : X → ℂ t : Tendsto (fun x => ↑(g x)) f (𝓝 ∞) ⊢ Tendsto (fun x => g x) f atInf TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.tendsto_inf_iff_tendsto_atInf
[318, 1]
[325, 41]
rw [←Function.comp_def, ←Filter.comap_comap, OnePoint.comap_coe_nhds_infty, Filter.coclosedCompact_eq_cocompact, ←atInf_eq_cocompact] at t
case mp X : Type f : Filter X g : X → ℂ t : f ≤ Filter.comap (fun x => ↑(g x)) (𝓝 ∞) ⊢ f ≤ Filter.comap (fun x => g x) atInf
case mp X : Type f : Filter X g : X → ℂ t : f ≤ Filter.comap (fun x => g x) atInf ⊢ f ≤ Filter.comap (fun x => g x) atInf
Please generate a tactic in lean4 to solve the state. STATE: case mp X : Type f : Filter X g : X → ℂ t : f ≤ Filter.comap (fun x => ↑(g x)) (𝓝 ∞) ⊢ f ≤ Filter.comap (fun x => g x) atInf TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.tendsto_inf_iff_tendsto_atInf
[318, 1]
[325, 41]
exact t
case mp X : Type f : Filter X g : X → ℂ t : f ≤ Filter.comap (fun x => g x) atInf ⊢ f ≤ Filter.comap (fun x => g x) atInf
no goals
Please generate a tactic in lean4 to solve the state. STATE: case mp X : Type f : Filter X g : X → ℂ t : f ≤ Filter.comap (fun x => g x) atInf ⊢ f ≤ Filter.comap (fun x => g x) atInf TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.tendsto_inf_iff_tendsto_atInf
[318, 1]
[325, 41]
exact fun h ↦ coe_tendsto_inf.comp h
case mpr X : Type f : Filter X g : X → ℂ ⊢ Tendsto (fun x => g x) f atInf → Tendsto (fun x => ↑(g x)) f (𝓝 ∞)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case mpr X : Type f : Filter X g : X → ℂ ⊢ Tendsto (fun x => g x) f atInf → Tendsto (fun x => ↑(g x)) f (𝓝 ∞) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
intro s o
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T ⊢ IsOpenMap fun z => ↑z
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s ⊢ IsOpen ((fun z => ↑z) '' s)
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T ⊢ IsOpenMap fun z => ↑z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
have e : (fun z : ℂ ↦ (z : 𝕊)) '' s = {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s := by apply Set.ext; intro z simp only [mem_image, mem_inter_iff, mem_compl_singleton_iff, mem_preimage] constructor intro ⟨x, m, e⟩; simp only [← e, toComplex_coe, m, and_true_iff]; exact inf_ne_coe.symm intro ⟨n, m⟩; use z.toComplex, m, coe_toComplex n
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s ⊢ IsOpen ((fun z => ↑z) '' s)
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s e : (fun z => ↑z) '' s = {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s ⊢ IsOpen ((fun z => ↑z) '' s)
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s ⊢ IsOpen ((fun z => ↑z) '' s) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
rw [e]
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s e : (fun z => ↑z) '' s = {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s ⊢ IsOpen ((fun z => ↑z) '' s)
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s e : (fun z => ↑z) '' s = {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s ⊢ IsOpen ({∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s)
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s e : (fun z => ↑z) '' s = {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s ⊢ IsOpen ((fun z => ↑z) '' s) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
exact continuousOn_toComplex.isOpen_inter_preimage isOpen_compl_singleton o
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s e : (fun z => ↑z) '' s = {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s ⊢ IsOpen ({∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s)
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s e : (fun z => ↑z) '' s = {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s ⊢ IsOpen ({∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
apply Set.ext
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s ⊢ (fun z => ↑z) '' s = {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s
case h X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s ⊢ ∀ (x : 𝕊), x ∈ (fun z => ↑z) '' s ↔ x ∈ {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s ⊢ (fun z => ↑z) '' s = {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
intro z
case h X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s ⊢ ∀ (x : 𝕊), x ∈ (fun z => ↑z) '' s ↔ x ∈ {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s
case h X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ∈ (fun z => ↑z) '' s ↔ z ∈ {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s ⊢ ∀ (x : 𝕊), x ∈ (fun z => ↑z) '' s ↔ x ∈ {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
simp only [mem_image, mem_inter_iff, mem_compl_singleton_iff, mem_preimage]
case h X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ∈ (fun z => ↑z) '' s ↔ z ∈ {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s
case h X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ (∃ x ∈ s, ↑x = z) ↔ z ≠ ∞ ∧ z.toComplex ∈ s
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ∈ (fun z => ↑z) '' s ↔ z ∈ {∞}ᶜ ∩ OnePoint.toComplex ⁻¹' s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
constructor
case h X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ (∃ x ∈ s, ↑x = z) ↔ z ≠ ∞ ∧ z.toComplex ∈ s
case h.mp X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ (∃ x ∈ s, ↑x = z) → z ≠ ∞ ∧ z.toComplex ∈ s case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ≠ ∞ ∧ z.toComplex ∈ s → ∃ x ∈ s, ↑x = z
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ (∃ x ∈ s, ↑x = z) ↔ z ≠ ∞ ∧ z.toComplex ∈ s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
intro ⟨x, m, e⟩
case h.mp X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ (∃ x ∈ s, ↑x = z) → z ≠ ∞ ∧ z.toComplex ∈ s case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ≠ ∞ ∧ z.toComplex ∈ s → ∃ x ∈ s, ↑x = z
case h.mp X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 x : ℂ m : x ∈ s e : ↑x = z ⊢ z ≠ ∞ ∧ z.toComplex ∈ s case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ≠ ∞ ∧ z.toComplex ∈ s → ∃ x ∈ s, ↑x = z
Please generate a tactic in lean4 to solve the state. STATE: case h.mp X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ (∃ x ∈ s, ↑x = z) → z ≠ ∞ ∧ z.toComplex ∈ s case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ≠ ∞ ∧ z.toComplex ∈ s → ∃ x ∈ s, ↑x = z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
simp only [← e, toComplex_coe, m, and_true_iff]
case h.mp X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 x : ℂ m : x ∈ s e : ↑x = z ⊢ z ≠ ∞ ∧ z.toComplex ∈ s case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ≠ ∞ ∧ z.toComplex ∈ s → ∃ x ∈ s, ↑x = z
case h.mp X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 x : ℂ m : x ∈ s e : ↑x = z ⊢ ↑x ≠ ∞ case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ≠ ∞ ∧ z.toComplex ∈ s → ∃ x ∈ s, ↑x = z
Please generate a tactic in lean4 to solve the state. STATE: case h.mp X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 x : ℂ m : x ∈ s e : ↑x = z ⊢ z ≠ ∞ ∧ z.toComplex ∈ s case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ≠ ∞ ∧ z.toComplex ∈ s → ∃ x ∈ s, ↑x = z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
exact inf_ne_coe.symm
case h.mp X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 x : ℂ m : x ∈ s e : ↑x = z ⊢ ↑x ≠ ∞ case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ≠ ∞ ∧ z.toComplex ∈ s → ∃ x ∈ s, ↑x = z
case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ≠ ∞ ∧ z.toComplex ∈ s → ∃ x ∈ s, ↑x = z
Please generate a tactic in lean4 to solve the state. STATE: case h.mp X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 x : ℂ m : x ∈ s e : ↑x = z ⊢ ↑x ≠ ∞ case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ≠ ∞ ∧ z.toComplex ∈ s → ∃ x ∈ s, ↑x = z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
intro ⟨n, m⟩
case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ≠ ∞ ∧ z.toComplex ∈ s → ∃ x ∈ s, ↑x = z
case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 n : z ≠ ∞ m : z.toComplex ∈ s ⊢ ∃ x ∈ s, ↑x = z
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 ⊢ z ≠ ∞ ∧ z.toComplex ∈ s → ∃ x ∈ s, ↑x = z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.isOpenMap_coe
[332, 1]
[340, 86]
use z.toComplex, m, coe_toComplex n
case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 n : z ≠ ∞ m : z.toComplex ∈ s ⊢ ∃ x ∈ s, ↑x = z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ o : IsOpen s z : 𝕊 n : z ≠ ∞ m : z.toComplex ∈ s ⊢ ∃ x ∈ s, ↑x = z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.prod_nhds_eq
[342, 1]
[345, 65]
refine le_antisymm ?_ (continuousAt_fst.prod (continuous_coe.continuousAt.comp continuousAt_snd))
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T x : X z : ℂ ⊢ 𝓝 (x, ↑z) = Filter.map (fun p => (p.1, ↑p.2)) (𝓝 (x, z))
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T x : X z : ℂ ⊢ 𝓝 (x, ↑z) ≤ Filter.map (fun p => (p.1, ↑p.2)) (𝓝 (x, z))
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T x : X z : ℂ ⊢ 𝓝 (x, ↑z) = Filter.map (fun p => (p.1, ↑p.2)) (𝓝 (x, z)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.prod_nhds_eq
[342, 1]
[345, 65]
apply IsOpenMap.nhds_le
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T x : X z : ℂ ⊢ 𝓝 (x, ↑z) ≤ Filter.map (fun p => (p.1, ↑p.2)) (𝓝 (x, z))
case hf X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T x : X z : ℂ ⊢ IsOpenMap fun p => (p.1, ↑p.2)
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T x : X z : ℂ ⊢ 𝓝 (x, ↑z) ≤ Filter.map (fun p => (p.1, ↑p.2)) (𝓝 (x, z)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.prod_nhds_eq
[342, 1]
[345, 65]
exact IsOpenMap.id.prod isOpenMap_coe
case hf X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T x : X z : ℂ ⊢ IsOpenMap fun p => (p.1, ↑p.2)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hf X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T x : X z : ℂ ⊢ IsOpenMap fun p => (p.1, ↑p.2) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.mem_inf_of_mem_atInf
[347, 1]
[351, 87]
simp only [OnePoint.nhds_infty_eq, Filter.mem_sup, Filter.coclosedCompact_eq_cocompact, ← atInf_eq_cocompact, Filter.mem_map]
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ f : s ∈ atInf ⊢ (fun z => ↑z) '' s ∪ {∞} ∈ 𝓝 ∞
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ f : s ∈ atInf ⊢ OnePoint.some ⁻¹' ((fun z => ↑z) '' s ∪ {∞}) ∈ atInf ∧ (fun z => ↑z) '' s ∪ {∞} ∈ pure ∞
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ f : s ∈ atInf ⊢ (fun z => ↑z) '' s ∪ {∞} ∈ 𝓝 ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.mem_inf_of_mem_atInf
[347, 1]
[351, 87]
exact ⟨Filter.mem_of_superset f fun _ m ↦ Or.inl (mem_image_of_mem _ m), Or.inr rfl⟩
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ f : s ∈ atInf ⊢ OnePoint.some ⁻¹' ((fun z => ↑z) '' s ∪ {∞}) ∈ atInf ∧ (fun z => ↑z) '' s ∪ {∞} ∈ pure ∞
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set ℂ f : s ∈ atInf ⊢ OnePoint.some ⁻¹' ((fun z => ↑z) '' s ∪ {∞}) ∈ atInf ∧ (fun z => ↑z) '' s ∪ {∞} ∈ pure ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.prod_mem_inf_of_mem_atInf
[353, 1]
[366, 13]
rcases Filter.mem_prod_iff.mp f with ⟨t, tx, u, ui, sub⟩
X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set (X × ℂ) x : X f : s ∈ (𝓝 x).prod atInf ⊢ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞} ∈ 𝓝 (x, ∞)
case intro.intro.intro.intro X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set (X × ℂ) x : X f : s ∈ (𝓝 x).prod atInf t : Set X tx : t ∈ 𝓝 x u : Set ℂ ui : u ∈ atInf sub : t ×ˢ u ⊆ s ⊢ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞} ∈ 𝓝 (x, ∞)
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set (X × ℂ) x : X f : s ∈ (𝓝 x).prod atInf ⊢ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞} ∈ 𝓝 (x, ∞) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.prod_mem_inf_of_mem_atInf
[353, 1]
[366, 13]
rw [nhds_prod_eq]
case intro.intro.intro.intro X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set (X × ℂ) x : X f : s ∈ (𝓝 x).prod atInf t : Set X tx : t ∈ 𝓝 x u : Set ℂ ui : u ∈ atInf sub : t ×ˢ u ⊆ s ⊢ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞} ∈ 𝓝 (x, ∞)
case intro.intro.intro.intro X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set (X × ℂ) x : X f : s ∈ (𝓝 x).prod atInf t : Set X tx : t ∈ 𝓝 x u : Set ℂ ui : u ∈ atInf sub : t ×ˢ u ⊆ s ⊢ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞} ∈ 𝓝 x ×ˢ 𝓝 ∞
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set (X × ℂ) x : X f : s ∈ (𝓝 x).prod atInf t : Set X tx : t ∈ 𝓝 x u : Set ℂ ui : u ∈ atInf sub : t ×ˢ u ⊆ s ⊢ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞} ∈ 𝓝 (x, ∞) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.prod_mem_inf_of_mem_atInf
[353, 1]
[366, 13]
refine Filter.mem_prod_iff.mpr ⟨t, tx, (fun z : ℂ ↦ (z : 𝕊)) '' u ∪ {∞}, mem_inf_of_mem_atInf ui, ?_⟩
case intro.intro.intro.intro X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set (X × ℂ) x : X f : s ∈ (𝓝 x).prod atInf t : Set X tx : t ∈ 𝓝 x u : Set ℂ ui : u ∈ atInf sub : t ×ˢ u ⊆ s ⊢ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞} ∈ 𝓝 x ×ˢ 𝓝 ∞
case intro.intro.intro.intro X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set (X × ℂ) x : X f : s ∈ (𝓝 x).prod atInf t : Set X tx : t ∈ 𝓝 x u : Set ℂ ui : u ∈ atInf sub : t ×ˢ u ⊆ s ⊢ t ×ˢ ((fun z => ↑z) '' u ∪ {∞}) ⊆ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞}
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set (X × ℂ) x : X f : s ∈ (𝓝 x).prod atInf t : Set X tx : t ∈ 𝓝 x u : Set ℂ ui : u ∈ atInf sub : t ×ˢ u ⊆ s ⊢ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞} ∈ 𝓝 x ×ˢ 𝓝 ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/AnalyticManifold/RiemannSphere.lean
RiemannSphere.prod_mem_inf_of_mem_atInf
[353, 1]
[366, 13]
intro ⟨y, z⟩ ⟨yt, m⟩
case intro.intro.intro.intro X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set (X × ℂ) x : X f : s ∈ (𝓝 x).prod atInf t : Set X tx : t ∈ 𝓝 x u : Set ℂ ui : u ∈ atInf sub : t ×ˢ u ⊆ s ⊢ t ×ˢ ((fun z => ↑z) '' u ∪ {∞}) ⊆ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞}
case intro.intro.intro.intro X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set (X × ℂ) x : X f : s ∈ (𝓝 x).prod atInf t : Set X tx : t ∈ 𝓝 x u : Set ℂ ui : u ∈ atInf sub : t ×ˢ u ⊆ s y : X z : 𝕊 yt : (y, z).1 ∈ t m : (y, z).2 ∈ (fun z => ↑z) '' u ∪ {∞} ⊢ (y, z) ∈ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞}
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro X : Type inst✝⁴ : TopologicalSpace X Y : Type inst✝³ : TopologicalSpace Y T : Type inst✝² : TopologicalSpace T inst✝¹ : ChartedSpace ℂ T inst✝ : AnalyticManifold I T s : Set (X × ℂ) x : X f : s ∈ (𝓝 x).prod atInf t : Set X tx : t ∈ 𝓝 x u : Set ℂ ui : u ∈ atInf sub : t ×ˢ u ⊆ s ⊢ t ×ˢ ((fun z => ↑z) '' u ∪ {∞}) ⊆ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞} TACTIC: