url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [mem_inter_iff, mem_preimage, extChartAt_prod, extChartAt_eq_refl,
PartialEquiv.prod_source, PartialEquiv.refl_source, mem_prod_eq, mem_univ, true_and_iff,
PartialEquiv.prod_coe, PartialEquiv.refl_coe, id, ← ht] at m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x ∈ t
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
⊢ f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x ∈ t
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have inv := i.he.right_inv m.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
⊢ f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
⊢ f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
⊢ f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [Cinv.g] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
⊢ f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
⊢ f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
⊢ f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | generalize hq : i.he.symm = q | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
⊢ f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
⊢ f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | rw [hq] at inv mf | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : ↑i.he (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | rw [Cinv.he, ContDiffAt.toPartialHomeomorph_coe i.ha.contDiffAt i.has_dhe le_top] at inv | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : ↑i.he (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : ↑i.he (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have q1 : (q (x.1, extChartAt I (f c z) x.2)).1 = x.1 := by simp only [← hq, i.inv_fst _ m.2] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [Cinv.h, Cinv.f', Prod.eq_iff_fst_eq_snd_eq, q1] at inv | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | nth_rw 2 [← PartialEquiv.left_inv _ m.1] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) =
↑(extChartAt I (f c z)).symm (↑(extChartAt I (f c z)) x.2) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | nth_rw 2 [← inv.2] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) =
↑(extChartAt I (f c z)).symm (↑(extChartAt I (f c z)) x.2) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) =
↑(extChartAt I (f c z)).symm
(↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2))) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) =
↑(extChartAt I (f c z)).symm (↑(extChartAt I (f c z)) x.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | refine (PartialEquiv.left_inv _ mf).symm | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) =
↑(extChartAt I (f c z)).symm
(↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) =
↑(extChartAt I (f c z)).symm
(↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | rw [← ht] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ IsOpen t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ IsOpen ((extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ IsOpen t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | exact (continuousOn_extChartAt _ _).isOpen_inter_preimage (isOpen_extChartAt_source _ _)
i.he.open_target | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ IsOpen ((extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ IsOpen ((extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have m := ContDiffAt.image_mem_toPartialHomeomorph_target i.ha.contDiffAt i.has_dhe le_top | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have e : i.h (c, i.z') = (c, extChartAt I (f c z) (f c z)) := by
simp only [Cinv.h, Cinv.z', Cinv.f', PartialEquiv.left_inv _ (mem_extChartAt_source _ _)] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
e : i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | rw [e] at m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
e : i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
e : i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
e : i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | exact m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
e : i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
e : i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [Cinv.h, Cinv.z', Cinv.f', PartialEquiv.left_inv _ (mem_extChartAt_source _ _)] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
⊢ i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
⊢ i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [m', mem_inter_iff, mem_preimage, mem_extChartAt_source, true_and_iff, ← ht,
extChartAt_prod, PartialEquiv.prod_coe, extChartAt_eq_refl, PartialEquiv.refl_coe, id,
PartialEquiv.prod_source, prod_mk_mem_set_prod_eq, PartialEquiv.refl_source, mem_univ] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
⊢ (c, f c z) ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
⊢ (c, f c z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | refine ContinuousAt.eventually_mem ?_ (extChartAt_source_mem_nhds' I ?_) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source | case refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2)) (c, f c z)
case refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ f (c, f c z).1 (↑(extChartAt I z).symm (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2) ∈
(extChartAt I (f c z)).source | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | apply i.fa.continuousAt.comp₂_of_eq continuousAt_fst | case refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2)) (c, f c z) | case refine_1.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (extChartAt I z).symm.1 (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z)
case refine_1.e
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ((c, f c z).1, (extChartAt I z).symm.1 (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2) = (c, z) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2)) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | refine ContinuousAt.comp ?_ ?_ | case refine_1.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (extChartAt I z).symm.1 (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z) | case refine_1.hh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (extChartAt I z).invFun (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2
case refine_1.hh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (extChartAt I z).symm.1 (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [i.inv_at] | case refine_1.hh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (extChartAt I z).invFun (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2 | case refine_1.hh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (extChartAt I z).invFun (↑(extChartAt I z) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (extChartAt I z).invFun (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | exact continuousAt_extChartAt_symm I _ | case refine_1.hh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (extChartAt I z).invFun (↑(extChartAt I z) z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (extChartAt I z).invFun (↑(extChartAt I z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | apply continuousAt_snd.comp | case refine_1.hh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z) | case refine_1.hh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => ↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | refine (PartialHomeomorph.continuousAt i.he.symm ?_).comp ?_ | case refine_1.hh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => ↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z) | case refine_1.hh.refine_2.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2) ∈ i.he.symm.source
case refine_1.hh.refine_2.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => ↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [m', (he i).symm_source] | case refine_1.hh.refine_2.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2) ∈ i.he.symm.source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2) ∈ i.he.symm.source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | apply continuousAt_fst.prod | case refine_1.hh.refine_2.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z) | case refine_1.hh.refine_2.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => ↑(extChartAt I (f c z)) x.2) (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | apply (continuousAt_extChartAt I _).comp_of_eq | case refine_1.hh.refine_2.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => ↑(extChartAt I (f c z)) x.2) (c, f c z) | case refine_1.hh.refine_2.refine_2.hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => x.2) (c, f c z)
case refine_1.hh.refine_2.refine_2.hy
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ (c, f c z).2 = f c z | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => ↑(extChartAt I (f c z)) x.2) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | exact continuousAt_snd | case refine_1.hh.refine_2.refine_2.hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => x.2) (c, f c z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2.refine_2.hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => x.2) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | rfl | case refine_1.hh.refine_2.refine_2.hy
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ (c, f c z).2 = f c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2.refine_2.hy
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ (c, f c z).2 = f c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [i.inv_at, PartialEquiv.left_inv _ (mem_extChartAt_source _ _),
PartialEquiv.invFun_as_coe] | case refine_1.e
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ((c, f c z).1, (extChartAt I z).symm.1 (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2) = (c, z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.e
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ((c, f c z).1, (extChartAt I z).symm.1 (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2) = (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [i.inv_at, PartialEquiv.left_inv _ (mem_extChartAt_source _ _)] | case refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ f (c, f c z).1 (↑(extChartAt I z).symm (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2) ∈
(extChartAt I (f c z)).source | case refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ f c z ∈ (extChartAt I (f c z)).source | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ f (c, f c z).1 (↑(extChartAt I z).symm (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2) ∈
(extChartAt I (f c z)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | apply mem_extChartAt_source | case refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ f c z ∈ (extChartAt I (f c z)).source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ f c z ∈ (extChartAt I (f c z)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [← hq, i.inv_fst _ m.2] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | apply AnalyticAt.holomorphicAt | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) (I.prod I) ↑i.he.symm (c, i.fz') | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) (I.prod I) ↑i.he.symm (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | have d : ContDiffAt ℂ ⊤ i.he.symm _ :=
ContDiffAt.to_localInverse i.ha.contDiffAt i.has_dhe le_top | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | have e : i.h (c, i.z') = (c, i.fz') := by
simp only [Cinv.h, Cinv.fz', Cinv.f']
simp only [Cinv.z', (extChartAt I z).left_inv (mem_extChartAt_source _ _)] | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
e : i.h (c, i.z') = (c, i.fz')
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | rw [e] at d | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
e : i.h (c, i.z') = (c, i.fz')
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ ↑i.he.symm (c, i.fz')
e : i.h (c, i.z') = (c, i.fz')
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
e : i.h (c, i.z') = (c, i.fz')
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | exact (contDiffAt_iff_analytic_at2 le_top).mp d | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ ↑i.he.symm (c, i.fz')
e : i.h (c, i.z') = (c, i.fz')
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ ↑i.he.symm (c, i.fz')
e : i.h (c, i.z') = (c, i.fz')
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | simp only [Cinv.h, Cinv.fz', Cinv.f'] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ i.h (c, i.z') = (c, i.fz') | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ (c, ↑(extChartAt I (f c z)) (f c (↑(extChartAt I z).symm i.z'))) = (c, ↑(extChartAt I (f c z)) (f c z)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ i.h (c, i.z') = (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | simp only [Cinv.z', (extChartAt I z).left_inv (mem_extChartAt_source _ _)] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ (c, ↑(extChartAt I (f c z)) (f c (↑(extChartAt I z).symm i.z'))) = (c, ↑(extChartAt I (f c z)) (f c z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ (c, ↑(extChartAt I (f c z)) (f c (↑(extChartAt I z).symm i.z'))) = (c, ↑(extChartAt I (f c z)) (f c z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.ga | [296, 1] | [303, 19] | apply (HolomorphicAt.extChartAt_symm (mem_extChartAt_target I z)).comp_of_eq | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (uncurry i.g) (c, f c z) | case gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (fun x => (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z)
case e
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2 = ↑(extChartAt I z) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (uncurry i.g) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.ga | [296, 1] | [303, 19] | refine holomorphicAt_snd.comp (i.he_symm_holomorphic.comp_of_eq ?_ ?_) | case gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (fun x => (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z) | case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun x => (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z)
case gh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2) = (c, i.fz') | Please generate a tactic in lean4 to solve the state.
STATE:
case gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (fun x => (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.ga | [296, 1] | [303, 19] | apply holomorphicAt_fst.prod | case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun x => (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z) | case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I (f c z)) x.2) (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun x => (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.ga | [296, 1] | [303, 19] | refine (HolomorphicAt.extChartAt ?_).comp holomorphicAt_snd | case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I (f c z)) x.2) (c, f c z) | case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (c, f c z).2 ∈ (extChartAt I (f c z)).source | Please generate a tactic in lean4 to solve the state.
STATE:
case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I (f c z)) x.2) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.ga | [296, 1] | [303, 19] | exact mem_extChartAt_source _ _ | case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (c, f c z).2 ∈ (extChartAt I (f c z)).source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (c, f c z).2 ∈ (extChartAt I (f c z)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.ga | [296, 1] | [303, 19] | rfl | case gh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2) = (c, i.fz') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case gh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2) = (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.ga | [296, 1] | [303, 19] | exact i.inv_at | case e
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2 = ↑(extChartAt I z) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | complex_inverse_fun | [311, 1] | [320, 41] | have i : ComplexInverseFun.Cinv f c z :=
{ fa
nc } | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
⊢ ∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, f c z) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2) ∧ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
i : ComplexInverseFun.Cinv f c z
⊢ ∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, f c z) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2) ∧ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
⊢ ∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, f c z) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2) ∧ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | complex_inverse_fun | [311, 1] | [320, 41] | use i.g, i.ga, i.left_inv, i.right_inv | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
i : ComplexInverseFun.Cinv f c z
⊢ ∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, f c z) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2) ∧ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
i : ComplexInverseFun.Cinv f c z
⊢ ∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, f c z) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2) ∧ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | complex_inverse_fun' | [324, 1] | [333, 60] | set f' : ℂ → S → T := fun _ z ↦ f z | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | complex_inverse_fun' | [324, 1] | [333, 60] | have fa' : HolomorphicAt II I (uncurry f') (0, z) := fa.comp_of_eq holomorphicAt_snd rfl | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
fa' : HolomorphicAt (I.prod I) I (uncurry f') (0, z)
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | complex_inverse_fun' | [324, 1] | [333, 60] | rcases complex_inverse_fun fa' nc with ⟨g, ga, gf, fg⟩ | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
fa' : HolomorphicAt (I.prod I) I (uncurry f') (0, z)
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
fa' : HolomorphicAt (I.prod I) I (uncurry f') (0, z)
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (0, f' 0 z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (0, z), g x.1 (f' x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (0, f' 0 z), f' x.1 (g x.1 x.2) = x.2
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
fa' : HolomorphicAt (I.prod I) I (uncurry f') (0, z)
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | complex_inverse_fun' | [324, 1] | [333, 60] | use g 0, ga.comp (holomorphicAt_const.prod holomorphicAt_id),
(continuousAt_const.prod continuousAt_id).eventually gf,
(continuousAt_const.prod continuousAt_id).eventually fg | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
fa' : HolomorphicAt (I.prod I) I (uncurry f') (0, z)
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (0, f' 0 z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (0, z), g x.1 (f' x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (0, f' 0 z), f' x.1 (g x.1 x.2) = x.2
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
fa' : HolomorphicAt (I.prod I) I (uncurry f') (0, z)
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (0, f' 0 z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (0, z), g x.1 (f' x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (0, f' 0 z), f' x.1 (g x.1 x.2) = x.2
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Set.Nonempty.left_invCoe | [21, 1] | [23, 76] | intro ⟨x, m⟩ | X : Type
s : Set X
ne : s.Nonempty
⊢ ∀ (x : ↑s), ne.invCoe ↑x = x | X : Type
s : Set X
ne : s.Nonempty
x : X
m : x ∈ s
⊢ ne.invCoe ↑⟨x, m⟩ = ⟨x, m⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s : Set X
ne : s.Nonempty
⊢ ∀ (x : ↑s), ne.invCoe ↑x = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Set.Nonempty.left_invCoe | [21, 1] | [23, 76] | simp only [Set.Nonempty.invCoe, Subtype.coe_mk, m, dif_pos] | X : Type
s : Set X
ne : s.Nonempty
x : X
m : x ∈ s
⊢ ne.invCoe ↑⟨x, m⟩ = ⟨x, m⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s : Set X
ne : s.Nonempty
x : X
m : x ∈ s
⊢ ne.invCoe ↑⟨x, m⟩ = ⟨x, m⟩
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Set.Nonempty.right_invCoe | [25, 1] | [27, 73] | intro x m | X : Type
s : Set X
ne : s.Nonempty
⊢ ∀ x ∈ s, ↑(ne.invCoe x) = x | X : Type
s : Set X
ne : s.Nonempty
x : X
m : x ∈ s
⊢ ↑(ne.invCoe x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s : Set X
ne : s.Nonempty
⊢ ∀ x ∈ s, ↑(ne.invCoe x) = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Set.Nonempty.right_invCoe | [25, 1] | [27, 73] | simp only [Set.Nonempty.invCoe, m, dif_pos, Subtype.coe_mk] | X : Type
s : Set X
ne : s.Nonempty
x : X
m : x ∈ s
⊢ ↑(ne.invCoe x) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s : Set X
ne : s.Nonempty
x : X
m : x ∈ s
⊢ ↑(ne.invCoe x) = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Set.Nonempty.continuousOn_invCoe | [29, 1] | [32, 53] | rw [embedding_subtype_val.continuousOn_iff] | X : Type
s : Set X
ne : s.Nonempty
inst✝ : TopologicalSpace X
⊢ ContinuousOn ne.invCoe s | X : Type
s : Set X
ne : s.Nonempty
inst✝ : TopologicalSpace X
⊢ ContinuousOn (Subtype.val ∘ ne.invCoe) s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s : Set X
ne : s.Nonempty
inst✝ : TopologicalSpace X
⊢ ContinuousOn ne.invCoe s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Set.Nonempty.continuousOn_invCoe | [29, 1] | [32, 53] | apply continuousOn_id.congr | X : Type
s : Set X
ne : s.Nonempty
inst✝ : TopologicalSpace X
⊢ ContinuousOn (Subtype.val ∘ ne.invCoe) s | X : Type
s : Set X
ne : s.Nonempty
inst✝ : TopologicalSpace X
⊢ EqOn (Subtype.val ∘ ne.invCoe) id s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s : Set X
ne : s.Nonempty
inst✝ : TopologicalSpace X
⊢ ContinuousOn (Subtype.val ∘ ne.invCoe) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Set.Nonempty.continuousOn_invCoe | [29, 1] | [32, 53] | intro x m | X : Type
s : Set X
ne : s.Nonempty
inst✝ : TopologicalSpace X
⊢ EqOn (Subtype.val ∘ ne.invCoe) id s | X : Type
s : Set X
ne : s.Nonempty
inst✝ : TopologicalSpace X
x : X
m : x ∈ s
⊢ (Subtype.val ∘ ne.invCoe) x = id x | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s : Set X
ne : s.Nonempty
inst✝ : TopologicalSpace X
⊢ EqOn (Subtype.val ∘ ne.invCoe) id s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Set.Nonempty.continuousOn_invCoe | [29, 1] | [32, 53] | simp only [Function.comp, ne.right_invCoe _ m, id] | X : Type
s : Set X
ne : s.Nonempty
inst✝ : TopologicalSpace X
x : X
m : x ∈ s
⊢ (Subtype.val ∘ ne.invCoe) x = id x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
s : Set X
ne : s.Nonempty
inst✝ : TopologicalSpace X
x : X
m : x ∈ s
⊢ (Subtype.val ∘ ne.invCoe) x = id x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | constructor | X : Type
inst✝ : TopologicalSpace X
s : Set X
⊢ TotallyDisconnectedSpace ↑s ↔ IsTotallyDisconnected s | case mp
X : Type
inst✝ : TopologicalSpace X
s : Set X
⊢ TotallyDisconnectedSpace ↑s → IsTotallyDisconnected s
case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
⊢ IsTotallyDisconnected s → TotallyDisconnectedSpace ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
s : Set X
⊢ TotallyDisconnectedSpace ↑s ↔ IsTotallyDisconnected s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | intro h | case mp
X : Type
inst✝ : TopologicalSpace X
s : Set X
⊢ TotallyDisconnectedSpace ↑s → IsTotallyDisconnected s | case mp
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
⊢ IsTotallyDisconnected s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝ : TopologicalSpace X
s : Set X
⊢ TotallyDisconnectedSpace ↑s → IsTotallyDisconnected s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | by_cases ne : s.Nonempty | case mp
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
⊢ IsTotallyDisconnected s | case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
⊢ IsTotallyDisconnected s
case neg
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : ¬s.Nonempty
⊢ IsTotallyDisconnected s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
⊢ IsTotallyDisconnected s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | intro t ts tc | case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
⊢ IsTotallyDisconnected s | case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
⊢ t.Subsingleton | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
⊢ IsTotallyDisconnected s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | set t' := ne.invCoe '' t | case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
⊢ t.Subsingleton | case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
⊢ t.Subsingleton | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
⊢ t.Subsingleton
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | have tc' : IsPreconnected t' := tc.image _ (ne.continuousOn_invCoe.mono ts) | case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
⊢ t.Subsingleton | case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
⊢ t.Subsingleton | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
⊢ t.Subsingleton
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | have q := h.isTotallyDisconnected_univ _ (subset_univ _) tc' | case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
⊢ t.Subsingleton | case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
⊢ t.Subsingleton | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
⊢ t.Subsingleton
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | rw [e] | case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
e : t = (fun x => ↑x) '' t'
⊢ t.Subsingleton | case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
e : t = (fun x => ↑x) '' t'
⊢ ((fun x => ↑x) '' t').Subsingleton | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
e : t = (fun x => ↑x) '' t'
⊢ t.Subsingleton
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | exact q.image _ | case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
e : t = (fun x => ↑x) '' t'
⊢ ((fun x => ↑x) '' t').Subsingleton | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
e : t = (fun x => ↑x) '' t'
⊢ ((fun x => ↑x) '' t').Subsingleton
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | apply Set.ext | X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
⊢ t = (fun x => ↑x) '' t' | case h
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
⊢ ∀ (x : X), x ∈ t ↔ x ∈ (fun x => ↑x) '' t' | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
⊢ t = (fun x => ↑x) '' t'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | intro x | case h
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
⊢ ∀ (x : X), x ∈ t ↔ x ∈ (fun x => ↑x) '' t' | case h
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
⊢ x ∈ t ↔ x ∈ (fun x => ↑x) '' t' | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
⊢ ∀ (x : X), x ∈ t ↔ x ∈ (fun x => ↑x) '' t'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | simp only [mem_image] | case h
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
⊢ x ∈ t ↔ x ∈ (fun x => ↑x) '' t' | case h
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
⊢ x ∈ t ↔ ∃ x_1 ∈ t', ↑x_1 = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
⊢ x ∈ t ↔ x ∈ (fun x => ↑x) '' t'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | constructor | case h
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
⊢ x ∈ t ↔ ∃ x_1 ∈ t', ↑x_1 = x | case h.mp
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
⊢ x ∈ t → ∃ x_1 ∈ t', ↑x_1 = x
case h.mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
⊢ (∃ x_1 ∈ t', ↑x_1 = x) → x ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
⊢ x ∈ t ↔ ∃ x_1 ∈ t', ↑x_1 = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | intro xt | case h.mp
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
⊢ x ∈ t → ∃ x_1 ∈ t', ↑x_1 = x | case h.mp
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ∃ x_1 ∈ t', ↑x_1 = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
⊢ x ∈ t → ∃ x_1 ∈ t', ↑x_1 = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | use ⟨x, ts xt⟩ | case h.mp
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ∃ x_1 ∈ t', ↑x_1 = x | case h
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ⟨x, ⋯⟩ ∈ t' ∧ ↑⟨x, ⋯⟩ = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ∃ x_1 ∈ t', ↑x_1 = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | refine ⟨⟨x,xt,?_⟩,?_⟩ | case h
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ⟨x, ⋯⟩ ∈ t' ∧ ↑⟨x, ⋯⟩ = x | case h.refine_1
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ne.invCoe x = ⟨x, ⋯⟩
case h.refine_2
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ↑⟨x, ⋯⟩ = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ⟨x, ⋯⟩ ∈ t' ∧ ↑⟨x, ⋯⟩ = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | simp only [Subtype.ext_iff, Subtype.coe_mk, ne.right_invCoe _ (ts xt)] | case h.refine_1
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ne.invCoe x = ⟨x, ⋯⟩
case h.refine_2
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ↑⟨x, ⋯⟩ = x | case h.refine_2
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ↑⟨x, ⋯⟩ = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ne.invCoe x = ⟨x, ⋯⟩
case h.refine_2
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ↑⟨x, ⋯⟩ = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | rw [Subtype.coe_mk] | case h.refine_2
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ↑⟨x, ⋯⟩ = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
xt : x ∈ t
⊢ ↑⟨x, ⋯⟩ = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | intro ⟨⟨y, ys⟩, ⟨z, zt, zy⟩, yx⟩ | case h.mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
⊢ (∃ x_1 ∈ t', ↑x_1 = x) → x ∈ t | case h.mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x y : X
ys : y ∈ s
z : X
zt : z ∈ t
zy : ne.invCoe z = ⟨y, ys⟩
yx : ↑⟨y, ys⟩ = x
⊢ x ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x : X
⊢ (∃ x_1 ∈ t', ↑x_1 = x) → x ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | simp only [Subtype.coe_mk, Subtype.ext_iff, ne.right_invCoe _ (ts zt)] at yx zy | case h.mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x y : X
ys : y ∈ s
z : X
zt : z ∈ t
zy : ne.invCoe z = ⟨y, ys⟩
yx : ↑⟨y, ys⟩ = x
⊢ x ∈ t | case h.mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x y : X
ys : y ∈ s
z : X
zt : z ∈ t
yx : y = x
zy : z = y
⊢ x ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x y : X
ys : y ∈ s
z : X
zt : z ∈ t
zy : ne.invCoe z = ⟨y, ys⟩
yx : ↑⟨y, ys⟩ = x
⊢ x ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | rw [← yx, ← zy] | case h.mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x y : X
ys : y ∈ s
z : X
zt : z ∈ t
yx : y = x
zy : z = y
⊢ x ∈ t | case h.mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x y : X
ys : y ∈ s
z : X
zt : z ∈ t
yx : y = x
zy : z = y
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x y : X
ys : y ∈ s
z : X
zt : z ∈ t
yx : y = x
zy : z = y
⊢ x ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | exact zt | case h.mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x y : X
ys : y ∈ s
z : X
zt : z ∈ t
yx : y = x
zy : z = y
⊢ z ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s.Nonempty
t : Set X
ts : t ⊆ s
tc : IsPreconnected t
t' : Set ↑s := ne.invCoe '' t
tc' : IsPreconnected t'
q : t'.Subsingleton
x y : X
ys : y ∈ s
z : X
zt : z ∈ t
yx : y = x
zy : z = y
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | simp only [not_nonempty_iff_eq_empty] at ne | case neg
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : ¬s.Nonempty
⊢ IsTotallyDisconnected s | case neg
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s = ∅
⊢ IsTotallyDisconnected s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : ¬s.Nonempty
⊢ IsTotallyDisconnected s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | rw [ne] | case neg
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s = ∅
⊢ IsTotallyDisconnected s | case neg
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s = ∅
⊢ IsTotallyDisconnected ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s = ∅
⊢ IsTotallyDisconnected s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | exact isTotallyDisconnected_empty | case neg
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s = ∅
⊢ IsTotallyDisconnected ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : TotallyDisconnectedSpace ↑s
ne : s = ∅
⊢ IsTotallyDisconnected ∅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | intro h | case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
⊢ IsTotallyDisconnected s → TotallyDisconnectedSpace ↑s | case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : IsTotallyDisconnected s
⊢ TotallyDisconnectedSpace ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
⊢ IsTotallyDisconnected s → TotallyDisconnectedSpace ↑s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | refine ⟨?_⟩ | case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : IsTotallyDisconnected s
⊢ TotallyDisconnectedSpace ↑s | case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : IsTotallyDisconnected s
⊢ IsTotallyDisconnected univ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : IsTotallyDisconnected s
⊢ TotallyDisconnectedSpace ↑s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | apply embedding_subtype_val.isTotallyDisconnected | case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : IsTotallyDisconnected s
⊢ IsTotallyDisconnected univ | case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : IsTotallyDisconnected s
⊢ IsTotallyDisconnected (Subtype.val '' univ) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : IsTotallyDisconnected s
⊢ IsTotallyDisconnected univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | rw [Subtype.coe_image_univ] | case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : IsTotallyDisconnected s
⊢ IsTotallyDisconnected (Subtype.val '' univ) | case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : IsTotallyDisconnected s
⊢ IsTotallyDisconnected s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : IsTotallyDisconnected s
⊢ IsTotallyDisconnected (Subtype.val '' univ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | isTotallyDisconnected_iff_totally_disconnected_subtype | [35, 1] | [55, 41] | exact h | case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : IsTotallyDisconnected s
⊢ IsTotallyDisconnected s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝ : TopologicalSpace X
s : Set X
h : IsTotallyDisconnected s
⊢ IsTotallyDisconnected s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | not_countable_Ioo | [58, 1] | [59, 101] | rw [← Cardinal.le_aleph0_iff_set_countable, not_le, Cardinal.mk_Ioo_real h] | a b : ℝ
h : a < b
⊢ ¬(Ioo a b).Countable | a b : ℝ
h : a < b
⊢ Cardinal.aleph0 < Cardinal.continuum | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
h : a < b
⊢ ¬(Ioo a b).Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | not_countable_Ioo | [58, 1] | [59, 101] | apply Cardinal.cantor | a b : ℝ
h : a < b
⊢ Cardinal.aleph0 < Cardinal.continuum | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
h : a < b
⊢ Cardinal.aleph0 < Cardinal.continuum
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | generalize hR : {r | ∃ x y : X, dist x y = r} = R | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
⊢ TotallyDisconnectedSpace X | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
⊢ TotallyDisconnectedSpace X | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
⊢ TotallyDisconnectedSpace X
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | have rc : R.Countable := by
have e : R = range (uncurry dist) := by
apply Set.ext; intro r; simp only [mem_setOf, mem_range, Prod.exists, uncurry, ← hR]; rfl
rw [e]; exact countable_range _ | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
⊢ TotallyDisconnectedSpace X | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
⊢ TotallyDisconnectedSpace X | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
⊢ TotallyDisconnectedSpace X
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | refine ⟨?_⟩ | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
⊢ TotallyDisconnectedSpace X | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
⊢ IsTotallyDisconnected univ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
⊢ TotallyDisconnectedSpace X
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | apply isTotallyDisconnected_of_isClopen_set | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
⊢ IsTotallyDisconnected univ | case hX
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
⊢ Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
⊢ IsTotallyDisconnected univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | intro x y xy | case hX
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
⊢ Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U | case hX
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : x ≠ y
⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U | Please generate a tactic in lean4 to solve the state.
STATE:
case hX
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
⊢ Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
TACTIC:
|
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