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 | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
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:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | simp only [mem_prod_eq, mem_image, mem_union, mem_singleton_iff, mem_univ, true_and_iff,
Prod.ext_iff] at 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
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 {∞} | 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 ∈ t
m : (∃ x ∈ u, ↑x = z) ∨ z = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = z) ∨ z = ∞ | 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
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 {∞}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | induction' z using OnePoint.rec with z | 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 ∈ t
m : (∃ x ∈ u, ↑x = z) ∨ z = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = z) ∨ z = ∞ | case intro.intro.intro.intro.h₁
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
yt : y ∈ t
m : (∃ x ∈ u, ↑x = ∞) ∨ ∞ = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = ∞) ∨ ∞ = ∞
case intro.intro.intro.intro.h₂
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
yt : y ∈ t
z : ℂ
m : (∃ x ∈ u, ↑x = ↑z) ∨ ↑z = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = ↑z) ∨ ↑z = ∞ | 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
y : X
z : 𝕊
yt : y ∈ t
m : (∃ x ∈ u, ↑x = z) ∨ z = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = z) ∨ z = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | simp only [eq_self_iff_true, or_true_iff] | case intro.intro.intro.intro.h₁
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
yt : y ∈ t
m : (∃ x ∈ u, ↑x = ∞) ∨ ∞ = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = ∞) ∨ ∞ = ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.h₁
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
yt : y ∈ t
m : (∃ x ∈ u, ↑x = ∞) ∨ ∞ = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = ∞) ∨ ∞ = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | simp only [coe_eq_inf_iff, or_false_iff, coe_eq_coe] at m ⊢ | case intro.intro.intro.intro.h₂
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
yt : y ∈ t
z : ℂ
m : (∃ x ∈ u, ↑x = ↑z) ∨ ↑z = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = ↑z) ∨ ↑z = ∞ | case intro.intro.intro.intro.h₂
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
yt : y ∈ t
z : ℂ
m : ∃ x ∈ u, x = z
⊢ ∃ x ∈ s, x.1 = y ∧ x.2 = z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.h₂
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
yt : y ∈ t
z : ℂ
m : (∃ x ∈ u, ↑x = ↑z) ∨ ↑z = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = ↑z) ∨ ↑z = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | rcases m with ⟨w, wu, wz⟩ | case intro.intro.intro.intro.h₂
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
yt : y ∈ t
z : ℂ
m : ∃ x ∈ u, x = z
⊢ ∃ x ∈ s, x.1 = y ∧ x.2 = z | case intro.intro.intro.intro.h₂.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
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ ∃ x ∈ s, x.1 = y ∧ x.2 = z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.h₂
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
yt : y ∈ t
z : ℂ
m : ∃ x ∈ u, x = z
⊢ ∃ x ∈ s, x.1 = y ∧ x.2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | refine ⟨⟨y, z⟩, sub (mk_mem_prod yt ?_), rfl, rfl⟩ | case intro.intro.intro.intro.h₂.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
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ ∃ x ∈ s, x.1 = y ∧ x.2 = z | case intro.intro.intro.intro.h₂.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
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ z ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.h₂.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
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ ∃ x ∈ s, x.1 = y ∧ x.2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | rw [← wz] | case intro.intro.intro.intro.h₂.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
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ z ∈ u | case intro.intro.intro.intro.h₂.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
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ w ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.h₂.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
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ z ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | exact wu | case intro.intro.intro.intro.h₂.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
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ w ∈ u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.h₂.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
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ w ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | rw [holomorphic_iff] | 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
⊢ Holomorphic I I 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
⊢ (Continuous fun z => ↑z) ∧
∀ (x : ℂ), AnalyticAt ℂ (↑(extChartAt I ↑x) ∘ (fun z => ↑z) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) 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
⊢ Holomorphic I I fun z => ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | use continuous_coe | 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
⊢ (Continuous fun z => ↑z) ∧
∀ (x : ℂ), AnalyticAt ℂ (↑(extChartAt I ↑x) ∘ (fun z => ↑z) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
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 : ℂ), AnalyticAt ℂ (↑(extChartAt I ↑x) ∘ (fun z => ↑z) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) 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
⊢ (Continuous fun z => ↑z) ∧
∀ (x : ℂ), AnalyticAt ℂ (↑(extChartAt I ↑x) ∘ (fun z => ↑z) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | intro z | case right
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 : ℂ), AnalyticAt ℂ (↑(extChartAt I ↑x) ∘ (fun z => ↑z) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
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
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I ↑z) ∘ (fun z => ↑z) ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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 : ℂ), AnalyticAt ℂ (↑(extChartAt I ↑x) ∘ (fun z => ↑z) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | simp only [extChartAt_coe, extChartAt_eq_refl, PartialEquiv.refl_symm, PartialEquiv.refl_coe,
Function.comp_id, id_eq, Function.comp, PartialEquiv.invFun_as_coe] | case right
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
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I ↑z) ∘ (fun z => ↑z) ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) z) | case right
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => ↑coePartialEquiv.symm ↑x) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I ↑z) ∘ (fun z => ↑z) ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | rw [← PartialEquiv.invFun_as_coe] | case right
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => ↑coePartialEquiv.symm ↑x) z | case right
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => coePartialEquiv.invFun ↑x) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => ↑coePartialEquiv.symm ↑x) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | simp only [coePartialEquiv, toComplex_coe] | case right
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => coePartialEquiv.invFun ↑x) z | case right
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => x) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => coePartialEquiv.invFun ↑x) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | apply analyticAt_id | case right
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => x) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => x) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_toComplex | [376, 1] | [380, 22] | rw [holomorphicAt_iff] | 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
z : ℂ
⊢ HolomorphicAt I I OnePoint.toComplex ↑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
z : ℂ
⊢ ContinuousAt OnePoint.toComplex ↑z ∧
AnalyticAt ℂ (↑(extChartAt I (↑z).toComplex) ∘ OnePoint.toComplex ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑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
z : ℂ
⊢ HolomorphicAt I I OnePoint.toComplex ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_toComplex | [376, 1] | [380, 22] | use continuousAt_toComplex | 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
z : ℂ
⊢ ContinuousAt OnePoint.toComplex ↑z ∧
AnalyticAt ℂ (↑(extChartAt I (↑z).toComplex) ∘ OnePoint.toComplex ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z) | case right
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
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I (↑z).toComplex) ∘ OnePoint.toComplex ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑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
z : ℂ
⊢ ContinuousAt OnePoint.toComplex ↑z ∧
AnalyticAt ℂ (↑(extChartAt I (↑z).toComplex) ∘ OnePoint.toComplex ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_toComplex | [376, 1] | [380, 22] | simp only [toComplex_coe, Function.comp, extChartAt_coe, extChartAt_eq_refl, PartialEquiv.refl_coe,
id, PartialEquiv.symm_symm, coePartialEquiv_apply, coePartialEquiv_symm_apply] | case right
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
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I (↑z).toComplex) ∘ OnePoint.toComplex ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z) | case right
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => x) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I (↑z).toComplex) ∘ OnePoint.toComplex ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_toComplex | [376, 1] | [380, 22] | apply analyticAt_id | case right
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => x) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => x) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | rw [holomorphic_iff] | 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
⊢ Holomorphic I I fun z => z⁻¹ | 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
⊢ (Continuous fun z => z⁻¹) ∧
∀ (x : 𝕊), AnalyticAt ℂ (↑(extChartAt I x⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) 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
⊢ Holomorphic I I fun z => z⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | use continuous_inv | 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
⊢ (Continuous fun z => z⁻¹) ∧
∀ (x : 𝕊), AnalyticAt ℂ (↑(extChartAt I x⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
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 : 𝕊), AnalyticAt ℂ (↑(extChartAt I x⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) 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
⊢ (Continuous fun z => z⁻¹) ∧
∀ (x : 𝕊), AnalyticAt ℂ (↑(extChartAt I x⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | intro z | case right
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 : 𝕊), AnalyticAt ℂ (↑(extChartAt I x⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
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
z : 𝕊
⊢ AnalyticAt ℂ (↑(extChartAt I z⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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 : 𝕊), AnalyticAt ℂ (↑(extChartAt I x⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | induction' z using OnePoint.rec with z | case right
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
z : 𝕊
⊢ AnalyticAt ℂ (↑(extChartAt I z⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) z) | case right.h₁
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
⊢ AnalyticAt ℂ (↑(extChartAt I ∞⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞)
case right.h₂
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
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I (↑z)⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
z : 𝕊
⊢ AnalyticAt ℂ (↑(extChartAt I z⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | simp only [inv_inf, extChartAt_inf, ← coe_zero, extChartAt_coe, Function.comp,
PartialEquiv.trans_apply, Equiv.toPartialEquiv_apply, invEquiv_apply, coePartialEquiv_symm_apply,
toComplex_coe, PartialEquiv.coe_trans_symm, PartialEquiv.symm_symm, coePartialEquiv_apply,
Equiv.toPartialEquiv_symm_apply, invEquiv_symm, inv_inv] | case right.h₁
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
⊢ AnalyticAt ℂ (↑(extChartAt I ∞⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞) | case right.h₁
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
⊢ AnalyticAt ℂ (fun x => x) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h₁
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
⊢ AnalyticAt ℂ (↑(extChartAt I ∞⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | apply analyticAt_id | case right.h₁
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
⊢ AnalyticAt ℂ (fun x => x) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h₁
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
⊢ AnalyticAt ℂ (fun x => x) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | simp only [extChartAt_coe, PartialEquiv.symm_symm, Function.comp, coePartialEquiv_apply,
coePartialEquiv_symm_apply, toComplex_coe] | case right.h₂
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
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I (↑z)⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z) | case right.h₂
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h₂
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
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I (↑z)⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | by_cases z0 : z = 0 | case right.h₂
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z | case pos
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
z : ℂ
z0 : z = 0
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z
case neg
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
z : ℂ
z0 : ¬z = 0
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h₂
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
z : ℂ
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | simp only [z0, coe_zero, extChartAt_inf, PartialEquiv.trans_apply, coePartialEquiv_symm_apply,
invEquiv_apply, Equiv.toPartialEquiv_apply, inv_zero', inv_inv, toComplex_coe] | case pos
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
z : ℂ
z0 : z = 0
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z | case pos
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
z : ℂ
z0 : z = 0
⊢ AnalyticAt ℂ (fun x => x) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
z : ℂ
z0 : z = 0
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | apply analyticAt_id | case pos
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
z : ℂ
z0 : z = 0
⊢ AnalyticAt ℂ (fun x => x) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
z : ℂ
z0 : z = 0
⊢ AnalyticAt ℂ (fun x => x) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | simp only [inv_coe z0, extChartAt_coe, coePartialEquiv_symm_apply] | case neg
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
z : ℂ
z0 : ¬z = 0
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z | case neg
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
z : ℂ
z0 : ¬z = 0
⊢ AnalyticAt ℂ (fun x => (↑x)⁻¹.toComplex) z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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
z : ℂ
z0 : ¬z = 0
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | refine ((analyticAt_id _ _).inv z0).congr ?_ | case neg
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
z : ℂ
z0 : ¬z = 0
⊢ AnalyticAt ℂ (fun x => (↑x)⁻¹.toComplex) z | case neg
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
z : ℂ
z0 : ¬z = 0
⊢ (𝓝 z).EventuallyEq (fun x => (id x)⁻¹) fun x => (↑x)⁻¹.toComplex | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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
z : ℂ
z0 : ¬z = 0
⊢ AnalyticAt ℂ (fun x => (↑x)⁻¹.toComplex) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | refine (continuousAt_id.eventually_ne z0).mp (eventually_of_forall fun w w0 ↦ ?_) | case neg
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
z : ℂ
z0 : ¬z = 0
⊢ (𝓝 z).EventuallyEq (fun x => (id x)⁻¹) fun x => (↑x)⁻¹.toComplex | case neg
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
z : ℂ
z0 : ¬z = 0
w : ℂ
w0 : id w ≠ 0
⊢ (fun x => (id x)⁻¹) w = (fun x => (↑x)⁻¹.toComplex) w | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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
z : ℂ
z0 : ¬z = 0
⊢ (𝓝 z).EventuallyEq (fun x => (id x)⁻¹) fun x => (↑x)⁻¹.toComplex
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | rw [id] at w0 | case neg
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
z : ℂ
z0 : ¬z = 0
w : ℂ
w0 : id w ≠ 0
⊢ (fun x => (id x)⁻¹) w = (fun x => (↑x)⁻¹.toComplex) w | case neg
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
z : ℂ
z0 : ¬z = 0
w : ℂ
w0 : w ≠ 0
⊢ (fun x => (id x)⁻¹) w = (fun x => (↑x)⁻¹.toComplex) w | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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
z : ℂ
z0 : ¬z = 0
w : ℂ
w0 : id w ≠ 0
⊢ (fun x => (id x)⁻¹) w = (fun x => (↑x)⁻¹.toComplex) w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | simp only [inv_coe w0, toComplex_coe, id] | case neg
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
z : ℂ
z0 : ¬z = 0
w : ℂ
w0 : w ≠ 0
⊢ (fun x => (id x)⁻¹) w = (fun x => (↑x)⁻¹.toComplex) w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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
z : ℂ
z0 : ¬z = 0
w : ℂ
w0 : w ≠ 0
⊢ (fun x => (id x)⁻¹) w = (fun x => (↑x)⁻¹.toComplex) w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_fill_coe | [426, 1] | [428, 69] | simp only [OnePoint.continuousAt_coe, Function.comp, fill_coe, fc] | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : ContinuousAt f z
⊢ ContinuousAt (fill f y) ↑z | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : ContinuousAt f z
⊢ ContinuousAt (fill f y) ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_fill_inf | [431, 1] | [434, 63] | simp only [OnePoint.continuousAt_infty', lift_inf, Filter.coclosedCompact_eq_cocompact, ←
atInf_eq_cocompact, Function.comp, fill_coe, fill_inf, fi] | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞ | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_fill | [437, 1] | [441, 48] | rw [continuous_iff_continuousAt] | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ Continuous (fill f y) | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ (x : 𝕊), ContinuousAt (fill f y) 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ Continuous (fill f y)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_fill | [437, 1] | [441, 48] | intro 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ (x : 𝕊), ContinuousAt (fill f y) x | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
z : 𝕊
⊢ ContinuousAt (fill f y) 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ (x : 𝕊), ContinuousAt (fill f y) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_fill | [437, 1] | [441, 48] | induction z using OnePoint.rec | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
z : 𝕊
⊢ ContinuousAt (fill f y) z | case h₁
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞
case h₂
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
x✝ : ℂ
⊢ ContinuousAt (fill f y) ↑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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
z : 𝕊
⊢ ContinuousAt (fill f y) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_fill | [437, 1] | [441, 48] | exact continuousAt_fill_inf fi | case h₁
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_fill | [437, 1] | [441, 48] | exact continuousAt_fill_coe fc.continuousAt | case h₂
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
x✝ : ℂ
⊢ ContinuousAt (fill f y) ↑x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
x✝ : ℂ
⊢ ContinuousAt (fill f y) ↑x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_coe | [444, 1] | [451, 28] | have e : (fun x : 𝕊 ↦ f x.toComplex) =ᶠ[𝓝 ↑z] fill f y := by
simp only [OnePoint.nhds_coe_eq, Filter.EventuallyEq, Filter.eventually_map, toComplex_coe,
fill_coe, eq_self_iff_true, Filter.eventually_true] | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
⊢ HolomorphicAt I I (fill f y) ↑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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ HolomorphicAt I I (fill f y) ↑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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
⊢ HolomorphicAt I I (fill f y) ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_coe | [444, 1] | [451, 28] | refine HolomorphicAt.congr ?_ 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ HolomorphicAt I I (fill f y) ↑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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ HolomorphicAt I I (fun x => f x.toComplex) ↑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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ HolomorphicAt I I (fill f y) ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_coe | [444, 1] | [451, 28] | refine fa.comp_of_eq holomorphicAt_toComplex ?_ | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ HolomorphicAt I I (fun x => f x.toComplex) ↑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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ (↑z).toComplex = 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ HolomorphicAt I I (fun x => f x.toComplex) ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_coe | [444, 1] | [451, 28] | simp only [toComplex_coe] | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ (↑z).toComplex = z | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ (↑z).toComplex = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_coe | [444, 1] | [451, 28] | simp only [OnePoint.nhds_coe_eq, Filter.EventuallyEq, Filter.eventually_map, toComplex_coe,
fill_coe, eq_self_iff_true, Filter.eventually_true] | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
⊢ (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y) | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
⊢ (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | rw [holomorphicAt_iff] | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ HolomorphicAt I I (fill f y) ∞ | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞ ∧
AnalyticAt ℂ (↑(extChartAt I (fill f y ∞)) ∘ fill f y ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞) | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ HolomorphicAt I I (fill f y) ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | use continuousAt_fill_inf fi | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞ ∧
AnalyticAt ℂ (↑(extChartAt I (fill f y ∞)) ∘ fill f y ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞) | case right
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (↑(extChartAt I (fill f y ∞)) ∘ fill f y ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞) | 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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞ ∧
AnalyticAt ℂ (↑(extChartAt I (fill f y ∞)) ∘ fill f y ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | simp only [Function.comp, extChartAt, PartialHomeomorph.extend, fill, rec_inf,
modelWithCornersSelf_partialEquiv, PartialEquiv.trans_refl, chartAt_inf,
PartialHomeomorph.symm_toPartialEquiv, PartialEquiv.symm_symm, PartialHomeomorph.toFun_eq_coe,
invCoePartialHomeomorph_apply, PartialHomeomorph.coe_coe_symm, invCoePartialHomeomorph_symm_apply,
inv_inf, toComplex_zero] | case right
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (↑(extChartAt I (fill f y ∞)) ∘ fill f y ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞) | case right
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (fun x => ↑(chartAt ℂ y) (OnePoint.rec y f (↑x)⁻¹)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (↑(extChartAt I (fill f y ∞)) ∘ fill f y ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | rw [e] | case right
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
e : (fun z => ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹)) = fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
⊢ AnalyticAt ℂ (fun x => ↑(chartAt ℂ y) (OnePoint.rec y f (↑x)⁻¹)) 0 | case right
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
e : (fun z => ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹)) = fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
e : (fun z => ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹)) = fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
⊢ AnalyticAt ℂ (fun x => ↑(chartAt ℂ y) (OnePoint.rec y f (↑x)⁻¹)) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | clear e | case right
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
e : (fun z => ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹)) = fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0 | case right
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
e : (fun z => ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹)) = fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | apply Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt | case right
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0 | case right.hd
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z
case right.hc
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0
TACTIC:
|
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