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/Dynamics/Multibrot/Isomorphism.lean | ray_surj | [171, 1] | [174, 38] | rwa [← ec] | case h.mp
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c e : 𝕊
m : e ∈ multibrotExt d
ec : e = c
⊢ c ∈ multibrotExt d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c e : 𝕊
m : e ∈ multibrotExt d
ec : e = c
⊢ c ∈ multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | ray_surj | [171, 1] | [174, 38] | intro m | case h.mpr
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
⊢ c ∈ multibrotExt d → ∃ x ∈ multibrotExt d, (ray d ∘ bottcher d) x = c | case h.mpr
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
⊢ ∃ x ∈ multibrotExt d, (ray d ∘ bottcher d) x = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
⊢ c ∈ multibrotExt d → ∃ x ∈ multibrotExt d, (ray d ∘ bottcher d) x = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | ray_surj | [171, 1] | [174, 38] | use c, m, ray_bottcher m | case h.mpr
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
⊢ ∃ x ∈ multibrotExt d, (ray d ∘ bottcher d) x = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
⊢ ∃ x ∈ multibrotExt d, (ray d ∘ bottcher d) x = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | have i : ∀ p : ℂ × S, p ∈ s → ComplexInverseFun.Cinv f p.1 p.2 := by
intro ⟨c, z⟩ m; exact
{ fa := fa _ m
nc := nc _ m } | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
⊢ ∃ g, HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
⊢ ∃ g, HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
⊢ ∃ g, HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | generalize hg : (fun c w ↦
if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
⊢ ∃ g, HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
⊢ ∃ g, HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
⊢ ∃ g, HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | have ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ q : ℂ × T in 𝓝 (p.1, f p.1 p.2),
g q.1 q.2 = (i p m).g q.1 q.2 := by
intro ⟨c, z⟩ m; simp only
have n := nontrivialHolomorphicAt_of_mfderiv_ne_zero (fa _ m).along_snd (nc _ m); simp only at n
simp only [n.nhds_eq_map_nhds_param (fa _ m), Filter.eventually_map]
apply (i _ m).left_inv.mp; apply (so.eventually_mem m).mp
apply eventually_of_forall; intro ⟨e, w⟩ wm gf
simp only at gf
simp only [left _ _ wm, gf] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
⊢ ∃ g, HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
⊢ ∃ g, HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
⊢ ∃ g, HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | use g | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
⊢ ∃ g, HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2 | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
⊢ HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
⊢ ∃ g, HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | constructor | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
⊢ HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2 | case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
⊢ HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s)
case h.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
⊢ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
⊢ HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | intro ⟨c, z⟩ m | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
⊢ ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
c : ℂ
z : S
m : (c, z) ∈ s
⊢ ComplexInverseFun.Cinv f (c, z).1 (c, z).2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
⊢ ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | exact
{ fa := fa _ m
nc := nc _ m } | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
c : ℂ
z : S
m : (c, z) ∈ s
⊢ ComplexInverseFun.Cinv f (c, z).1 (c, z).2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
c : ℂ
z : S
m : (c, z) ∈ s
⊢ ComplexInverseFun.Cinv f (c, z).1 (c, z).2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | intro c z m | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
⊢ ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
⊢ g c (f c z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
⊢ ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | have h : ∃ x, (c, x) ∈ s ∧ f c x = f c z := ⟨z, m, rfl⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
⊢ g c (f c z) = z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
⊢ g c (f c z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
⊢ g c (f c z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | simp only [← hg, dif_pos h] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
⊢ g c (f c z) = z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
⊢ choose h = z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
⊢ g c (f c z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | rcases choose_spec h with ⟨m0, w0⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
⊢ choose h = z | case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
⊢ choose h = z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
⊢ choose h = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | have left := (i _ m).left_inv.self_of_nhds | case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
⊢ choose h = z | case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g ((c, z).1, (c, z).2).1 (f ((c, z).1, (c, z).2).1 ((c, z).1, (c, z).2).2) = ((c, z).1, (c, z).2).2
⊢ choose h = z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
⊢ choose h = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | simp only at left | case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g ((c, z).1, (c, z).2).1 (f ((c, z).1, (c, z).2).1 ((c, z).1, (c, z).2).2) = ((c, z).1, (c, z).2).2
⊢ choose h = z | case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
⊢ choose h = z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g ((c, z).1, (c, z).2).1 (f ((c, z).1, (c, z).2).1 ((c, z).1, (c, z).2).2) = ((c, z).1, (c, z).2).2
⊢ choose h = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | rw [left] at e | case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
e : (c, choose h) = (c, ⋯.g c (f c z))
⊢ choose h = z | case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
e : (c, choose h) = (c, z)
⊢ choose h = z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
e : (c, choose h) = (c, ⋯.g c (f c z))
⊢ choose h = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | exact (Prod.ext_iff.mp e).2 | case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
e : (c, choose h) = (c, z)
⊢ choose h = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
e : (c, choose h) = (c, z)
⊢ choose h = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | refine (inj.eq_iff m0 ?_).mp ?_ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
⊢ (c, choose h) = (c, ⋯.g c (f c z)) | case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
⊢ (c, ⋯.g c (f c z)) ∈ s
case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
⊢ ((c, choose h).1, f (c, choose h).1 (c, choose h).2) =
((c, ⋯.g c (f c z)).1, f (c, ⋯.g c (f c z)).1 (c, ⋯.g c (f c z)).2) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
⊢ (c, choose h) = (c, ⋯.g c (f c z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | simp only [left, m] | case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
⊢ (c, ⋯.g c (f c z)) ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
⊢ (c, ⋯.g c (f c z)) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | simp only [left, w0] | case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
⊢ ((c, choose h).1, f (c, choose h).1 (c, choose h).2) =
((c, ⋯.g c (f c z)).1, f (c, ⋯.g c (f c z)).1 (c, ⋯.g c (f c z)).2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
c : ℂ
z : S
m : (c, z) ∈ s
h : ∃ x, (c, x) ∈ s ∧ f c x = f c z
m0 : (c, choose h) ∈ s
w0 : f c (choose h) = f c z
left : ⋯.g c (f c z) = z
⊢ ((c, choose h).1, f (c, choose h).1 (c, choose h).2) =
((c, ⋯.g c (f c z)).1, f (c, ⋯.g c (f c z)).1 (c, ⋯.g c (f c z)).2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | intro ⟨c, z⟩ m | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
⊢ ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
⊢ ∀ᶠ (q : ℂ × T) in 𝓝 ((c, z).1, f (c, z).1 (c, z).2), g q.1 q.2 = ⋯.g q.1 q.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
⊢ ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | simp only | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
⊢ ∀ᶠ (q : ℂ × T) in 𝓝 ((c, z).1, f (c, z).1 (c, z).2), g q.1 q.2 = ⋯.g q.1 q.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
⊢ ∀ᶠ (q : ℂ × T) in 𝓝 (c, f c z), g q.1 q.2 = ⋯.g q.1 q.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
⊢ ∀ᶠ (q : ℂ × T) in 𝓝 ((c, z).1, f (c, z).1 (c, z).2), g q.1 q.2 = ⋯.g q.1 q.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | have n := nontrivialHolomorphicAt_of_mfderiv_ne_zero (fa _ m).along_snd (nc _ m) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
⊢ ∀ᶠ (q : ℂ × T) in 𝓝 (c, f c z), g q.1 q.2 = ⋯.g q.1 q.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ᶠ (q : ℂ × T) in 𝓝 (c, f c z), g q.1 q.2 = ⋯.g q.1 q.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
⊢ ∀ᶠ (q : ℂ × T) in 𝓝 (c, f c z), g q.1 q.2 = ⋯.g q.1 q.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | simp only [n.nhds_eq_map_nhds_param (fa _ m), Filter.eventually_map] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ᶠ (q : ℂ × T) in 𝓝 (c, f c z), g q.1 q.2 = ⋯.g q.1 q.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ᶠ (a : ℂ × S) in 𝓝 (c, z), g a.1 (f a.1 a.2) = ⋯.g a.1 (f a.1 a.2) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ᶠ (q : ℂ × T) in 𝓝 (c, f c z), g q.1 q.2 = ⋯.g q.1 q.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | apply (i _ m).left_inv.mp | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ᶠ (a : ℂ × S) in 𝓝 (c, z), g a.1 (f a.1 a.2) = ⋯.g a.1 (f a.1 a.2) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 ((c, z).1, (c, z).2), ⋯.g x.1 (f x.1 x.2) = x.2 → g x.1 (f x.1 x.2) = ⋯.g x.1 (f x.1 x.2) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ᶠ (a : ℂ × S) in 𝓝 (c, z), g a.1 (f a.1 a.2) = ⋯.g a.1 (f a.1 a.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | apply (so.eventually_mem m).mp | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 ((c, z).1, (c, z).2), ⋯.g x.1 (f x.1 x.2) = x.2 → g x.1 (f x.1 x.2) = ⋯.g x.1 (f x.1 x.2) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), x ∈ s → ⋯.g x.1 (f x.1 x.2) = x.2 → g x.1 (f x.1 x.2) = ⋯.g x.1 (f x.1 x.2) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 ((c, z).1, (c, z).2), ⋯.g x.1 (f x.1 x.2) = x.2 → g x.1 (f x.1 x.2) = ⋯.g x.1 (f x.1 x.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | apply eventually_of_forall | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), x ∈ s → ⋯.g x.1 (f x.1 x.2) = x.2 → g x.1 (f x.1 x.2) = ⋯.g x.1 (f x.1 x.2) | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ x ∈ s, ⋯.g x.1 (f x.1 x.2) = x.2 → g x.1 (f x.1 x.2) = ⋯.g x.1 (f x.1 x.2) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), x ∈ s → ⋯.g x.1 (f x.1 x.2) = x.2 → g x.1 (f x.1 x.2) = ⋯.g x.1 (f x.1 x.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | intro ⟨e, w⟩ wm gf | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ x ∈ s, ⋯.g x.1 (f x.1 x.2) = x.2 → g x.1 (f x.1 x.2) = ⋯.g x.1 (f x.1 x.2) | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
e : ℂ
w : S
wm : (e, w) ∈ s
gf : ⋯.g (e, w).1 (f (e, w).1 (e, w).2) = (e, w).2
⊢ g (e, w).1 (f (e, w).1 (e, w).2) = ⋯.g (e, w).1 (f (e, w).1 (e, w).2) | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
⊢ ∀ x ∈ s, ⋯.g x.1 (f x.1 x.2) = x.2 → g x.1 (f x.1 x.2) = ⋯.g x.1 (f x.1 x.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | simp only at gf | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
e : ℂ
w : S
wm : (e, w) ∈ s
gf : ⋯.g (e, w).1 (f (e, w).1 (e, w).2) = (e, w).2
⊢ g (e, w).1 (f (e, w).1 (e, w).2) = ⋯.g (e, w).1 (f (e, w).1 (e, w).2) | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
e : ℂ
w : S
wm : (e, w) ∈ s
gf : ⋯.g e (f e w) = w
⊢ g (e, w).1 (f (e, w).1 (e, w).2) = ⋯.g (e, w).1 (f (e, w).1 (e, w).2) | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
e : ℂ
w : S
wm : (e, w) ∈ s
gf : ⋯.g (e, w).1 (f (e, w).1 (e, w).2) = (e, w).2
⊢ g (e, w).1 (f (e, w).1 (e, w).2) = ⋯.g (e, w).1 (f (e, w).1 (e, w).2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | simp only [left _ _ wm, gf] | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
e : ℂ
w : S
wm : (e, w) ∈ s
gf : ⋯.g e (f e w) = w
⊢ g (e, w).1 (f (e, w).1 (e, w).2) = ⋯.g (e, w).1 (f (e, w).1 (e, w).2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
c : ℂ
z : S
m : (c, z) ∈ s
n : NontrivialHolomorphicAt (fun y => f c y) z
e : ℂ
w : S
wm : (e, w) ∈ s
gf : ⋯.g e (f e w) = w
⊢ g (e, w).1 (f (e, w).1 (e, w).2) = ⋯.g (e, w).1 (f (e, w).1 (e, w).2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | intro ⟨c, w⟩ wm | case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
⊢ HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) | case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
⊢ HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | rcases(mem_image _ _ _).mp wm with ⟨⟨c', z⟩, zm, e⟩ | case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, w) | case h.left.intro.mk.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
c' : ℂ
z : S
zm : (c', z) ∈ s
e : ((c', z).1, f (c', z).1 (c', z).2) = (c, w)
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | simp only [Prod.ext_iff] at e | case h.left.intro.mk.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
c' : ℂ
z : S
zm : (c', z) ∈ s
e : ((c', z).1, f (c', z).1 (c', z).2) = (c, w)
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, w) | case h.left.intro.mk.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
c' : ℂ
z : S
zm : (c', z) ∈ s
e : c' = c ∧ f c' z = w
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.intro.mk.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
c' : ℂ
z : S
zm : (c', z) ∈ s
e : ((c', z).1, f (c', z).1 (c', z).2) = (c, w)
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | simp only [e.1] at e zm | case h.left.intro.mk.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
c' : ℂ
z : S
zm : (c', z) ∈ s
e : c' = c ∧ f c' z = w
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, w) | case h.left.intro.mk.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
c' : ℂ
z : S
e : True ∧ f c z = w
zm : (c, z) ∈ s
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.intro.mk.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
c' : ℂ
z : S
zm : (c', z) ∈ s
e : c' = c ∧ f c' z = w
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | simp only [← e.2] | case h.left.intro.mk.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
c' : ℂ
z : S
e : True ∧ f c z = w
zm : (c, z) ∈ s
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, w) | case h.left.intro.mk.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
c' : ℂ
z : S
e : True ∧ f c z = w
zm : (c, z) ∈ s
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.intro.mk.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
c' : ℂ
z : S
e : True ∧ f c z = w
zm : (c, z) ∈ s
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | exact (i _ zm).ga.congr (Filter.EventuallyEq.symm (ge _ zm)) | case h.left.intro.mk.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
c' : ℂ
z : S
e : True ∧ f c z = w
zm : (c, z) ∈ s
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, f c z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.intro.mk.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
w : T
wm : (c, w) ∈ (fun p => (p.1, f p.1 p.2)) '' s
c' : ℂ
z : S
e : True ∧ f c z = w
zm : (c, z) ∈ s
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | intro ⟨c, z⟩ m | case h.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
⊢ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2 | case h.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
z : S
m : (c, z) ∈ s
⊢ g (c, z).1 (f (c, z).1 (c, z).2) = (c, z).2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
⊢ ∀ p ∈ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_open | [32, 1] | [70, 37] | exact left _ _ m | case h.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
z : S
m : (c, z) ∈ s
⊢ g (c, z).1 (f (c, z).1 (c, z).2) = (c, z).2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝ : Nonempty S
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
so : IsOpen s
i : ∀ p ∈ s, ComplexInverseFun.Cinv f p.1 p.2
g : ℂ → T → S
hg : (fun c w => if h : ∃ z, (c, z) ∈ s ∧ f c z = w then choose h else Classical.arbitrary S) = g
left : ∀ (c : ℂ) (z : S), (c, z) ∈ s → g c (f c z) = z
ge : ∀ (p : ℂ × S) (m : p ∈ s), ∀ᶠ (q : ℂ × T) in 𝓝 (p.1, f p.1 p.2), g q.1 q.2 = ⋯.g q.1 q.2
c : ℂ
z : S
m : (c, z) ∈ s
⊢ g (c, z).1 (f (c, z).1 (c, z).2) = (c, z).2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | have t : ∃ t, IsOpen t ∧ s ⊆ t ∧ InjOn (fun p : ℂ × S ↦ (p.1, f p.1 p.2)) t := by
apply inj.exists_isOpen_superset sc (fun _ m ↦ continuousAt_fst.prod (fa _ m).continuousAt)
intro ⟨c, z⟩ m; rcases complex_inverse_fun (fa _ m) (nc _ m) with ⟨g, _, gf, _⟩
rcases eventually_nhds_iff.mp gf with ⟨t, gf, o, m⟩
use t, o.mem_nhds m; intro ⟨c0, z0⟩ m0 ⟨c1, z1⟩ m1 e
simp only [uncurry, Prod.ext_iff] at e ⊢; use e.1
have e0 := gf _ m0; have e1 := gf _ m1; simp only at e0 e1
rw [← e0, ← e1, e.2, ← e.1] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : ∃ t, IsOpen t ∧ s ⊆ t ∧ InjOn (fun p => (p.1, f p.1 p.2)) t
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | rcases t with ⟨t, ot, st, ti⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : ∃ t, IsOpen t ∧ s ⊆ t ∧ InjOn (fun p => (p.1, f p.1 p.2)) t
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : ∃ t, IsOpen t ∧ s ⊆ t ∧ InjOn (fun p => (p.1, f p.1 p.2)) t
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | set u := t ∩ {p | HolomorphicAt II I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0} | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | have tu : u ⊆ t := inter_subset_left t _ | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | have su : s ⊆ u := subset_inter st (subset_inter fa nc) | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | have uo : IsOpen u := by
apply ot.inter; rw [isOpen_iff_eventually]; intro ⟨c, z⟩ ⟨fa, nc⟩
refine fa.eventually.mp ((mfderiv_ne_zero_eventually' fa nc).mp (eventually_of_forall ?_))
intro ⟨c, z⟩ nc fa; use fa, nc | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
uo : IsOpen u
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | have fa' : HolomorphicOn II I (uncurry f) u := fun _ m ↦ (inter_subset_right _ _ m).1 | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
uo : IsOpen u
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
uo : IsOpen u
fa' : HolomorphicOn (I.prod I) I (uncurry f) u
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
uo : IsOpen u
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | have d0 : ∀ (p : ℂ × S), p ∈ u → mfderiv I I (f p.fst) p.snd ≠ 0 :=
fun _ m ↦ (inter_subset_right _ _ m).2 | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
uo : IsOpen u
fa' : HolomorphicOn (I.prod I) I (uncurry f) u
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
uo : IsOpen u
fa' : HolomorphicOn (I.prod I) I (uncurry f) u
d0 : ∀ p ∈ u, mfderiv I I (f p.1) p.2 ≠ 0
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
uo : IsOpen u
fa' : HolomorphicOn (I.prod I) I (uncurry f) u
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | rcases global_complex_inverse_fun_open fa' d0 (ti.mono tu) uo with ⟨g, ga, gf⟩ | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
uo : IsOpen u
fa' : HolomorphicOn (I.prod I) I (uncurry f) u
d0 : ∀ p ∈ u, mfderiv I I (f p.1) p.2 ≠ 0
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | case intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
uo : IsOpen u
fa' : HolomorphicOn (I.prod I) I (uncurry f) u
d0 : ∀ p ∈ u, mfderiv I I (f p.1) p.2 ≠ 0
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' u)
gf : ∀ p ∈ u, g p.1 (f p.1 p.2) = p.2
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
uo : IsOpen u
fa' : HolomorphicOn (I.prod I) I (uncurry f) u
d0 : ∀ p ∈ u, mfderiv I I (f p.1) p.2 ≠ 0
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | use g, ga.mono (image_subset _ su), Filter.eventually_of_mem (uo.mem_nhdsSet.mpr su) gf | case intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
uo : IsOpen u
fa' : HolomorphicOn (I.prod I) I (uncurry f) u
d0 : ∀ p ∈ u, mfderiv I I (f p.1) p.2 ≠ 0
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' u)
gf : ∀ p ∈ u, g p.1 (f p.1 p.2) = p.2
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
uo : IsOpen u
fa' : HolomorphicOn (I.prod I) I (uncurry f) u
d0 : ∀ p ∈ u, mfderiv I I (f p.1) p.2 ≠ 0
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' u)
gf : ∀ p ∈ u, g p.1 (f p.1 p.2) = p.2
⊢ ∃ g,
HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f p.1 p.2)) '' s) ∧
∀ᶠ (p : ℂ × S) in 𝓝ˢ s, g p.1 (f p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | apply inj.exists_isOpen_superset sc (fun _ m ↦ continuousAt_fst.prod (fa _ m).continuousAt) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
⊢ ∃ t, IsOpen t ∧ s ⊆ t ∧ InjOn (fun p => (p.1, f p.1 p.2)) t | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
⊢ ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn (fun p => (p.1, f p.1 p.2)) u | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
⊢ ∃ t, IsOpen t ∧ s ⊆ t ∧ InjOn (fun p => (p.1, f p.1 p.2)) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | intro ⟨c, z⟩ m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
⊢ ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn (fun p => (p.1, f p.1 p.2)) u | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m : (c, z) ∈ s
⊢ ∃ u ∈ 𝓝 (c, z), InjOn (fun p => (p.1, f p.1 p.2)) u | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
⊢ ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn (fun p => (p.1, f p.1 p.2)) u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | rcases complex_inverse_fun (fa _ m) (nc _ m) with ⟨g, _, gf, _⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m : (c, z) ∈ s
⊢ ∃ u ∈ 𝓝 (c, z), InjOn (fun p => (p.1, f p.1 p.2)) u | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
⊢ ∃ u ∈ 𝓝 (c, z), InjOn (fun p => (p.1, f p.1 p.2)) u | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m : (c, z) ∈ s
⊢ ∃ u ∈ 𝓝 (c, z), InjOn (fun p => (p.1, f p.1 p.2)) u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | rcases eventually_nhds_iff.mp gf with ⟨t, gf, o, m⟩ | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
⊢ ∃ u ∈ 𝓝 (c, z), InjOn (fun p => (p.1, f p.1 p.2)) u | case intro.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
⊢ ∃ u ∈ 𝓝 (c, z), InjOn (fun p => (p.1, f p.1 p.2)) u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
⊢ ∃ u ∈ 𝓝 (c, z), InjOn (fun p => (p.1, f p.1 p.2)) u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | use t, o.mem_nhds m | case intro.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
⊢ ∃ u ∈ 𝓝 (c, z), InjOn (fun p => (p.1, f p.1 p.2)) u | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
⊢ InjOn (fun p => (p.1, f p.1 p.2)) t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
⊢ ∃ u ∈ 𝓝 (c, z), InjOn (fun p => (p.1, f p.1 p.2)) u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | intro ⟨c0, z0⟩ m0 ⟨c1, z1⟩ m1 e | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
⊢ InjOn (fun p => (p.1, f p.1 p.2)) t | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : (fun p => (p.1, f p.1 p.2)) (c0, z0) = (fun p => (p.1, f p.1 p.2)) (c1, z1)
⊢ (c0, z0) = (c1, z1) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
⊢ InjOn (fun p => (p.1, f p.1 p.2)) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | simp only [uncurry, Prod.ext_iff] at e ⊢ | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : (fun p => (p.1, f p.1 p.2)) (c0, z0) = (fun p => (p.1, f p.1 p.2)) (c1, z1)
⊢ (c0, z0) = (c1, z1) | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
⊢ c0 = c1 ∧ z0 = z1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : (fun p => (p.1, f p.1 p.2)) (c0, z0) = (fun p => (p.1, f p.1 p.2)) (c1, z1)
⊢ (c0, z0) = (c1, z1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | use e.1 | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
⊢ c0 = c1 ∧ z0 = z1 | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
⊢ z0 = z1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
⊢ c0 = c1 ∧ z0 = z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | have e0 := gf _ m0 | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
⊢ z0 = z1 | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
e0 : g (c0, z0).1 (f (c0, z0).1 (c0, z0).2) = (c0, z0).2
⊢ z0 = z1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
⊢ z0 = z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | have e1 := gf _ m1 | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
e0 : g (c0, z0).1 (f (c0, z0).1 (c0, z0).2) = (c0, z0).2
⊢ z0 = z1 | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
e0 : g (c0, z0).1 (f (c0, z0).1 (c0, z0).2) = (c0, z0).2
e1 : g (c1, z1).1 (f (c1, z1).1 (c1, z1).2) = (c1, z1).2
⊢ z0 = z1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
e0 : g (c0, z0).1 (f (c0, z0).1 (c0, z0).2) = (c0, z0).2
⊢ z0 = z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | simp only at e0 e1 | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
e0 : g (c0, z0).1 (f (c0, z0).1 (c0, z0).2) = (c0, z0).2
e1 : g (c1, z1).1 (f (c1, z1).1 (c1, z1).2) = (c1, z1).2
⊢ z0 = z1 | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
e0 : g c0 (f c0 z0) = z0
e1 : g c1 (f c1 z1) = z1
⊢ z0 = z1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
e0 : g (c0, z0).1 (f (c0, z0).1 (c0, z0).2) = (c0, z0).2
e1 : g (c1, z1).1 (f (c1, z1).1 (c1, z1).2) = (c1, z1).2
⊢ z0 = z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | rw [← e0, ← e1, e.2, ← e.1] | case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
e0 : g c0 (f c0 z0) = z0
e1 : g c1 (f c1 z1) = z1
⊢ z0 = z1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
c : ℂ
z : S
m✝ : (c, z) ∈ s
g : ℂ → T → S
left✝ : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf✝ : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
right✝ : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
t : Set (ℂ × S)
gf : ∀ x ∈ t, g x.1 (f x.1 x.2) = x.2
o : IsOpen t
m : (c, z) ∈ t
c0 : ℂ
z0 : S
m0 : (c0, z0) ∈ t
c1 : ℂ
z1 : S
m1 : (c1, z1) ∈ t
e : c0 = c1 ∧ f c0 z0 = f c1 z1
e0 : g c0 (f c0 z0) = z0
e1 : g c1 (f c1 z1) = z1
⊢ z0 = z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | apply ot.inter | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
⊢ IsOpen u | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
⊢ IsOpen {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0} | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
⊢ IsOpen u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | rw [isOpen_iff_eventually] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
⊢ IsOpen {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0} | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
⊢ ∀ x ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0},
∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0} | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
⊢ IsOpen {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | intro ⟨c, z⟩ ⟨fa, nc⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
⊢ ∀ x ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0},
∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0} | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa✝ : HolomorphicOn (I.prod I) I (uncurry f) s
nc✝ : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f (c, z).1) (c, z).2 ≠ 0
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0} | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa : HolomorphicOn (I.prod I) I (uncurry f) s
nc : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
⊢ ∀ x ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0},
∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | refine fa.eventually.mp ((mfderiv_ne_zero_eventually' fa nc).mp (eventually_of_forall ?_)) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa✝ : HolomorphicOn (I.prod I) I (uncurry f) s
nc✝ : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f (c, z).1) (c, z).2 ≠ 0
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0} | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa✝ : HolomorphicOn (I.prod I) I (uncurry f) s
nc✝ : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f (c, z).1) (c, z).2 ≠ 0
⊢ ∀ (x : ℂ × S),
mfderiv I I (f x.1) x.2 ≠ 0 →
HolomorphicAt (I.prod I) I (uncurry f) x →
x ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0} | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa✝ : HolomorphicOn (I.prod I) I (uncurry f) s
nc✝ : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f (c, z).1) (c, z).2 ≠ 0
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | intro ⟨c, z⟩ nc fa | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa✝ : HolomorphicOn (I.prod I) I (uncurry f) s
nc✝ : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f (c, z).1) (c, z).2 ≠ 0
⊢ ∀ (x : ℂ × S),
mfderiv I I (f x.1) x.2 ≠ 0 →
HolomorphicAt (I.prod I) I (uncurry f) x →
x ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0} | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa✝¹ : HolomorphicOn (I.prod I) I (uncurry f) s
nc✝¹ : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
c✝ : ℂ
z✝ : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c✝, z✝)
nc✝ : mfderiv I I (f (c✝, z✝).1) (c✝, z✝).2 ≠ 0
c : ℂ
z : S
nc : mfderiv I I (f (c, z).1) (c, z).2 ≠ 0
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ (c, z) ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0} | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa✝ : HolomorphicOn (I.prod I) I (uncurry f) s
nc✝ : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f (c, z).1) (c, z).2 ≠ 0
⊢ ∀ (x : ℂ × S),
mfderiv I I (f x.1) x.2 ≠ 0 →
HolomorphicAt (I.prod I) I (uncurry f) x →
x ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | global_complex_inverse_fun_compact | [74, 1] | [104, 90] | use fa, nc | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa✝¹ : HolomorphicOn (I.prod I) I (uncurry f) s
nc✝¹ : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
c✝ : ℂ
z✝ : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c✝, z✝)
nc✝ : mfderiv I I (f (c✝, z✝).1) (c✝, z✝).2 ≠ 0
c : ℂ
z : S
nc : mfderiv I I (f (c, z).1) (c, z).2 ≠ 0
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ (c, z) ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
inst✝¹ : Nonempty S
inst✝ : T2Space T
s : Set (ℂ × S)
fa✝¹ : HolomorphicOn (I.prod I) I (uncurry f) s
nc✝¹ : ∀ p ∈ s, mfderiv I I (f p.1) p.2 ≠ 0
inj : InjOn (fun p => (p.1, f p.1 p.2)) s
sc : IsCompact s
t : Set (ℂ × S)
ot : IsOpen t
st : s ⊆ t
ti : InjOn (fun p => (p.1, f p.1 p.2)) t
u : Set (ℂ × S) := t ∩ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
tu : u ⊆ t
su : s ⊆ u
c✝ : ℂ
z✝ : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c✝, z✝)
nc✝ : mfderiv I I (f (c✝, z✝).1) (c✝, z✝).2 ≠ 0
c : ℂ
z : S
nc : mfderiv I I (f (c, z).1) (c, z).2 ≠ 0
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ (c, z) ∈ {p | HolomorphicAt (I.prod I) I (uncurry f) p ∧ mfderiv I I (f p.1) p.2 ≠ 0}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | set f' := fun (_ : ℂ) (z : S) ↦ f z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | have nc' : ∀ p : ℂ × S, p ∈ (univ : Set ℂ) ×ˢ s → mfderiv I I (f' p.1) p.2 ≠ 0 := by
intro ⟨c, z⟩ ⟨_, zs⟩; exact nc _ zs | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | have inj' : InjOn (fun p : ℂ × S ↦ (p.1, f' p.1 p.2)) (univ ×ˢ s) := by
intro ⟨c0, z0⟩ ⟨_, zs0⟩ ⟨c1, z1⟩ ⟨_, zs1⟩ h; simp only [Prod.ext_iff] at h zs0 zs1
rw [h.1, inj zs0 zs1]; exact h.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | have fa' : HolomorphicOn II I (uncurry f') (univ ×ˢ s) := by
intro ⟨c, z⟩ ⟨_, zs⟩; exact (fa z zs).comp_of_eq holomorphicAt_snd rfl | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | rcases global_complex_inverse_fun_open fa' nc' inj' (isOpen_univ.prod so) with ⟨g, ga, gf⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z | case intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | use g 0 | case intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
⊢ HolomorphicOn I I (g 0) (f '' s) ∧ ∀ z ∈ s, g 0 (f z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
⊢ ∃ g, HolomorphicOn I I g (f '' s) ∧ ∀ z ∈ s, g (f z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | constructor | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
⊢ HolomorphicOn I I (g 0) (f '' s) ∧ ∀ z ∈ s, g 0 (f z) = z | case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
⊢ HolomorphicOn I I (g 0) (f '' s)
case h.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
⊢ ∀ z ∈ s, g 0 (f z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
⊢ HolomorphicOn I I (g 0) (f '' s) ∧ ∀ z ∈ s, g 0 (f z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | intro ⟨c, z⟩ ⟨_, zs⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
⊢ ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
c : ℂ
z : S
left✝ : (c, z).1 ∈ univ
zs : (c, z).2 ∈ s
⊢ mfderiv I I (f' (c, z).1) (c, z).2 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
⊢ ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | exact nc _ zs | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
c : ℂ
z : S
left✝ : (c, z).1 ∈ univ
zs : (c, z).2 ∈ s
⊢ mfderiv I I (f' (c, z).1) (c, z).2 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
c : ℂ
z : S
left✝ : (c, z).1 ∈ univ
zs : (c, z).2 ∈ s
⊢ mfderiv I I (f' (c, z).1) (c, z).2 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | intro ⟨c0, z0⟩ ⟨_, zs0⟩ ⟨c1, z1⟩ ⟨_, zs1⟩ h | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
⊢ InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
c0 : ℂ
z0 : S
left✝¹ : (c0, z0).1 ∈ univ
zs0 : (c0, z0).2 ∈ s
c1 : ℂ
z1 : S
left✝ : (c1, z1).1 ∈ univ
zs1 : (c1, z1).2 ∈ s
h : (fun p => (p.1, f' p.1 p.2)) (c0, z0) = (fun p => (p.1, f' p.1 p.2)) (c1, z1)
⊢ (c0, z0) = (c1, z1) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
⊢ InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | simp only [Prod.ext_iff] at h zs0 zs1 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
c0 : ℂ
z0 : S
left✝¹ : (c0, z0).1 ∈ univ
zs0 : (c0, z0).2 ∈ s
c1 : ℂ
z1 : S
left✝ : (c1, z1).1 ∈ univ
zs1 : (c1, z1).2 ∈ s
h : (fun p => (p.1, f' p.1 p.2)) (c0, z0) = (fun p => (p.1, f' p.1 p.2)) (c1, z1)
⊢ (c0, z0) = (c1, z1) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
c0 : ℂ
z0 : S
left✝¹ : (c0, z0).1 ∈ univ
zs0 : z0 ∈ s
c1 : ℂ
z1 : S
left✝ : (c1, z1).1 ∈ univ
zs1 : z1 ∈ s
h : c0 = c1 ∧ f' c0 z0 = f' c1 z1
⊢ (c0, z0) = (c1, z1) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
c0 : ℂ
z0 : S
left✝¹ : (c0, z0).1 ∈ univ
zs0 : (c0, z0).2 ∈ s
c1 : ℂ
z1 : S
left✝ : (c1, z1).1 ∈ univ
zs1 : (c1, z1).2 ∈ s
h : (fun p => (p.1, f' p.1 p.2)) (c0, z0) = (fun p => (p.1, f' p.1 p.2)) (c1, z1)
⊢ (c0, z0) = (c1, z1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | rw [h.1, inj zs0 zs1] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
c0 : ℂ
z0 : S
left✝¹ : (c0, z0).1 ∈ univ
zs0 : z0 ∈ s
c1 : ℂ
z1 : S
left✝ : (c1, z1).1 ∈ univ
zs1 : z1 ∈ s
h : c0 = c1 ∧ f' c0 z0 = f' c1 z1
⊢ (c0, z0) = (c1, z1) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
c0 : ℂ
z0 : S
left✝¹ : (c0, z0).1 ∈ univ
zs0 : z0 ∈ s
c1 : ℂ
z1 : S
left✝ : (c1, z1).1 ∈ univ
zs1 : z1 ∈ s
h : c0 = c1 ∧ f' c0 z0 = f' c1 z1
⊢ f z0 = f z1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
c0 : ℂ
z0 : S
left✝¹ : (c0, z0).1 ∈ univ
zs0 : z0 ∈ s
c1 : ℂ
z1 : S
left✝ : (c1, z1).1 ∈ univ
zs1 : z1 ∈ s
h : c0 = c1 ∧ f' c0 z0 = f' c1 z1
⊢ (c0, z0) = (c1, z1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | exact h.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
c0 : ℂ
z0 : S
left✝¹ : (c0, z0).1 ∈ univ
zs0 : z0 ∈ s
c1 : ℂ
z1 : S
left✝ : (c1, z1).1 ∈ univ
zs1 : z1 ∈ s
h : c0 = c1 ∧ f' c0 z0 = f' c1 z1
⊢ f z0 = f z1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
c0 : ℂ
z0 : S
left✝¹ : (c0, z0).1 ∈ univ
zs0 : z0 ∈ s
c1 : ℂ
z1 : S
left✝ : (c1, z1).1 ∈ univ
zs1 : z1 ∈ s
h : c0 = c1 ∧ f' c0 z0 = f' c1 z1
⊢ f z0 = f z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | intro ⟨c, z⟩ ⟨_, zs⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
⊢ HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
c : ℂ
z : S
left✝ : (c, z).1 ∈ univ
zs : (c, z).2 ∈ s
⊢ HolomorphicAt (I.prod I) I (uncurry f') (c, z) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
⊢ HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | exact (fa z zs).comp_of_eq holomorphicAt_snd rfl | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
c : ℂ
z : S
left✝ : (c, z).1 ∈ univ
zs : (c, z).2 ∈ s
⊢ HolomorphicAt (I.prod I) I (uncurry f') (c, z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
c : ℂ
z : S
left✝ : (c, z).1 ∈ univ
zs : (c, z).2 ∈ s
⊢ HolomorphicAt (I.prod I) I (uncurry f') (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | intro z ⟨w, m⟩ | case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
⊢ HolomorphicOn I I (g 0) (f '' s) | case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
z : T
w : S
m : w ∈ s ∧ f w = z
⊢ HolomorphicAt I I (g 0) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
⊢ HolomorphicOn I I (g 0) (f '' s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | refine (ga ⟨0, z⟩ ⟨⟨0, w⟩, ⟨mem_univ _, m.1⟩, ?_⟩).along_snd | case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
z : T
w : S
m : w ∈ s ∧ f w = z
⊢ HolomorphicAt I I (g 0) z | case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
z : T
w : S
m : w ∈ s ∧ f w = z
⊢ (fun p => (p.1, f' p.1 p.2)) (0, w) = (0, z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
z : T
w : S
m : w ∈ s ∧ f w = z
⊢ HolomorphicAt I I (g 0) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | simp only [Prod.ext_iff, eq_self_iff_true, true_and_iff] | case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
z : T
w : S
m : w ∈ s ∧ f w = z
⊢ (fun p => (p.1, f' p.1 p.2)) (0, w) = (0, z) | case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
z : T
w : S
m : w ∈ s ∧ f w = z
⊢ f' 0 w = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
z : T
w : S
m : w ∈ s ∧ f w = z
⊢ (fun p => (p.1, f' p.1 p.2)) (0, w) = (0, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | exact m.2 | case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
z : T
w : S
m : w ∈ s ∧ f w = z
⊢ f' 0 w = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
z : T
w : S
m : w ∈ s ∧ f w = z
⊢ f' 0 w = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | intro z m | case h.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
⊢ ∀ z ∈ s, g 0 (f z) = z | case h.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
z : S
m : z ∈ s
⊢ g 0 (f z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
⊢ ∀ z ∈ s, g 0 (f z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/GlobalInverse.lean | weak_global_complex_inverse_fun_open | [109, 1] | [124, 47] | exact gf ⟨0, z⟩ ⟨mem_univ _, m⟩ | case h.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
z : S
m : z ∈ s
⊢ g 0 (f z) = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
inst✝ : Nonempty S
s : Set S
fa : HolomorphicOn I I f s
nc : ∀ z ∈ s, mfderiv I I f z ≠ 0
inj : InjOn f s
so : IsOpen s
f' : ℂ → S → T := fun x z => f z
nc' : ∀ p ∈ univ ×ˢ s, mfderiv I I (f' p.1) p.2 ≠ 0
inj' : InjOn (fun p => (p.1, f' p.1 p.2)) (univ ×ˢ s)
fa' : HolomorphicOn (I.prod I) I (uncurry f') (univ ×ˢ s)
g : ℂ → T → S
ga : HolomorphicOn (I.prod I) I (uncurry g) ((fun p => (p.1, f' p.1 p.2)) '' univ ×ˢ s)
gf : ∀ p ∈ univ ×ˢ s, g p.1 (f' p.1 p.2) = p.2
z : S
m : z ∈ s
⊢ g 0 (f z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Duals.lean | dualVector_le | [47, 1] | [52, 38] | rw [← Complex.norm_eq_abs] | G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
x y : E
⊢ Complex.abs ((dualVector x) y) ≤ ‖y‖ | G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
x y : E
⊢ ‖(dualVector x) y‖ ≤ ‖y‖ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
x y : E
⊢ Complex.abs ((dualVector x) y) ≤ ‖y‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Duals.lean | dualVector_le | [47, 1] | [52, 38] | calc ‖dualVector x y‖
_ ≤ ‖dualVector x‖ * ‖y‖ := (dualVector x).le_op_norm y
_ ≤ 1 * ‖y‖ := by bound
_ = ‖y‖ := by simp only [one_mul] | G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
x y : E
⊢ ‖(dualVector x) y‖ ≤ ‖y‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
x y : E
⊢ ‖(dualVector x) y‖ ≤ ‖y‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Duals.lean | dualVector_le | [47, 1] | [52, 38] | bound | G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
x y : E
⊢ ‖dualVector x‖ * ‖y‖ ≤ 1 * ‖y‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
x y : E
⊢ ‖dualVector x‖ * ‖y‖ ≤ 1 * ‖y‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Duals.lean | dualVector_le | [47, 1] | [52, 38] | simp only [one_mul] | G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
x y : E
⊢ 1 * ‖y‖ = ‖y‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
x y : E
⊢ 1 * ‖y‖ = ‖y‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Duals.lean | LipschitzWith.is_const | [58, 1] | [59, 101] | intro x y | G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
g0 : LipschitzWith 0 g
⊢ ∀ (x y : ℝ), g x = g y | G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
g0 : LipschitzWith 0 g
x y : ℝ
⊢ g x = g y | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
g0 : LipschitzWith 0 g
⊢ ∀ (x y : ℝ), g x = g y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Duals.lean | LipschitzWith.is_const | [58, 1] | [59, 101] | simpa only [ENNReal.coe_zero, zero_mul, nonpos_iff_eq_zero, edist_eq_zero] using g0 x y | G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
g0 : LipschitzWith 0 g
x y : ℝ
⊢ g x = g y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
g0 : LipschitzWith 0 g
x y : ℝ
⊢ g x = g y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Duals.lean | duals_bddAbove | [62, 1] | [67, 54] | rw [bddAbove_def] | G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
⊢ BddAbove (range fun n => g ‖(duals n) x‖) | G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
⊢ ∃ x_1, ∀ y ∈ range fun n => g ‖(duals n) x‖, y ≤ x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
⊢ BddAbove (range fun n => g ‖(duals n) x‖)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Duals.lean | duals_bddAbove | [62, 1] | [67, 54] | use g ‖x‖ | G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
⊢ ∃ x_1, ∀ y ∈ range fun n => g ‖(duals n) x‖, y ≤ x_1 | case h
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
⊢ ∀ y ∈ range fun n => g ‖(duals n) x‖, y ≤ g ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
⊢ ∃ x_1, ∀ y ∈ range fun n => g ‖(duals n) x‖, y ≤ x_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Duals.lean | duals_bddAbove | [62, 1] | [67, 54] | simp only [Complex.norm_eq_abs, Set.mem_range, forall_exists_index,
forall_apply_eq_imp_iff'] | case h
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
⊢ ∀ y ∈ range fun n => g ‖(duals n) x‖, y ≤ g ‖x‖ | case h
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
⊢ ∀ (y : ℝ) (x_1 : ℕ), g (Complex.abs ((duals x_1) x)) = y → y ≤ g ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
⊢ ∀ y ∈ range fun n => g ‖(duals n) x‖, y ≤ g ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Duals.lean | duals_bddAbove | [62, 1] | [67, 54] | intro _ _ h | case h
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
⊢ ∀ (y : ℝ) (x_1 : ℕ), g (Complex.abs ((duals x_1) x)) = y → y ≤ g ‖x‖ | case h
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
y✝ : ℝ
x✝ : ℕ
h : g (Complex.abs ((duals x✝) x)) = y✝
⊢ y✝ ≤ g ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
⊢ ∀ (y : ℝ) (x_1 : ℕ), g (Complex.abs ((duals x_1) x)) = y → y ≤ g ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Duals.lean | duals_bddAbove | [62, 1] | [67, 54] | rw [←h] | case h
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
y✝ : ℝ
x✝ : ℕ
h : g (Complex.abs ((duals x✝) x)) = y✝
⊢ y✝ ≤ g ‖x‖ | case h
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
y✝ : ℝ
x✝ : ℕ
h : g (Complex.abs ((duals x✝) x)) = y✝
⊢ g (Complex.abs ((duals x✝) x)) ≤ g ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
y✝ : ℝ
x✝ : ℕ
h : g (Complex.abs ((duals x✝) x)) = y✝
⊢ y✝ ≤ g ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Duals.lean | duals_bddAbove | [62, 1] | [67, 54] | apply gm | case h
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
y✝ : ℝ
x✝ : ℕ
h : g (Complex.abs ((duals x✝) x)) = y✝
⊢ g (Complex.abs ((duals x✝) x)) ≤ g ‖x‖ | case h.a
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
y✝ : ℝ
x✝ : ℕ
h : g (Complex.abs ((duals x✝) x)) = y✝
⊢ Complex.abs ((duals x✝) x) ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝³ : NormedAddCommGroup G
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : SecondCountableTopology E
g : ℝ → ℝ
gm : Monotone g
x : E
y✝ : ℝ
x✝ : ℕ
h : g (Complex.abs ((duals x✝) x)) = y✝
⊢ g (Complex.abs ((duals x✝) x)) ≤ g ‖x‖
TACTIC:
|
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