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
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stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
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stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | exact ff | case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
c : ℂ
f' : ℂ → ℂ := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ℂ → ℂ
ga : AnalyticAt ℂ g 0
e : g 0 = 0
h✝ : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → g x ≠ x ∧ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (𝓝 c) (𝓝 0)
z : ℂ
h : z ≠ c → g (z - c) ≠ z - c ∧ f' (g (z - c)) = f' (z - c)
zc : z ≠ c
gz : g (z - c) ≠ z - c
ff : f (g (z - c) + c) = f z
⊢ f (g (z - c) + c) = f z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
c : ℂ
f' : ℂ → ℂ := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ℂ → ℂ
ga : AnalyticAt ℂ g 0
e : g 0 = 0
h✝ : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → g x ≠ x ∧ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (𝓝 c) (𝓝 0)
z : ℂ
h : z ≠ c → g (z - c) ≠ z - c ∧ f' (g (z - c)) = f' (z - c)
zc : z ≠ c
gz : g (z - c) ≠ z - c
ff : f (g (z - c) + c) = f z
⊢ f (g (z - c) + c) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | rw [← sub_self c] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
c : ℂ
f' : ℂ → ℂ := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ℂ → ℂ
ga : AnalyticAt ℂ g 0
e : g 0 = 0
h : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → g x ≠ x ∧ f' (g x) = f' x
⊢ Tendsto (fun z => z - c) (𝓝 c) (𝓝 0) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
c : ℂ
f' : ℂ → ℂ := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ℂ → ℂ
ga : AnalyticAt ℂ g 0
e : g 0 = 0
h : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → g x ≠ x ∧ f' (g x) = f' x
⊢ Tendsto (fun z => z - c) (𝓝 c) (𝓝 (c - c)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
c : ℂ
f' : ℂ → ℂ := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ℂ → ℂ
ga : AnalyticAt ℂ g 0
e : g 0 = 0
h : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → g x ≠ x ∧ f' (g x) = f' x
⊢ Tendsto (fun z => z - c) (𝓝 c) (𝓝 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | exact continuousAt_id.sub continuousAt_const | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
c : ℂ
f' : ℂ → ℂ := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ℂ → ℂ
ga : AnalyticAt ℂ g 0
e : g 0 = 0
h : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → g x ≠ x ∧ f' (g x) = f' x
⊢ Tendsto (fun z => z - c) (𝓝 c) (𝓝 (c - c)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
c : ℂ
f' : ℂ → ℂ := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ℂ → ℂ
ga : AnalyticAt ℂ g 0
e : g 0 = 0
h : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → g x ≠ x ∧ f' (g x) = f' x
⊢ Tendsto (fun z => z - c) (𝓝 c) (𝓝 (c - c))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | generalize hg : (fun z ↦ extChartAt I (f c) (f ((extChartAt I c).symm z))) = g | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | have dg : mfderiv I I g (extChartAt I c c) = 0 := by
have fd : MDifferentiableAt I I f ((extChartAt I c).symm (extChartAt I c c)) := by
rw [PartialEquiv.left_inv]; exact fa.mdifferentiableAt; apply mem_extChartAt_source
rw [← hg, ←Function.comp_def, mfderiv_comp _ (HolomorphicAt.extChartAt _).mdifferentiableAt _,
←Function.comp_def, mfderiv_comp _ fd (HolomorphicAt.extChartAt_symm _).mdifferentiableAt,
PartialEquiv.left_inv, df, ContinuousLinearMap.zero_comp, ContinuousLinearMap.comp_zero]
apply mem_extChartAt_source; apply mem_extChartAt_target; rw [PartialEquiv.left_inv]
apply mem_extChartAt_source; apply mem_extChartAt_source
exact MDifferentiableAt.comp _ fd
(HolomorphicAt.extChartAt_symm (mem_extChartAt_target _ _)).mdifferentiableAt | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
dg : mfderiv I I g (↑(extChartAt I c) c) = 0
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | simp only [holomorphicAt_iff, Function.comp, hg] at fa | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
dg : mfderiv I I g (↑(extChartAt I c) c) = 0
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
dg : mfderiv I I g (↑(extChartAt I c) c) = 0
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
dg : mfderiv I I g (↑(extChartAt I c) c) = 0
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | have dg' := fa.2.differentiableAt.mdifferentiableAt.hasMFDerivAt | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
dg : mfderiv I I g (↑(extChartAt I c) c) = 0
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
dg : mfderiv I I g (↑(extChartAt I c) c) = 0
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg' : HasMFDerivAt I I g (↑(extChartAt I c) c) (mfderiv I I g (↑(extChartAt I c) c))
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
dg : mfderiv I I g (↑(extChartAt I c) c) = 0
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [dg, hasMFDerivAt_iff_hasFDerivAt] at dg' | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
dg : mfderiv I I g (↑(extChartAt I c) c) = 0
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg' : HasMFDerivAt I I g (↑(extChartAt I c) c) (mfderiv I I g (↑(extChartAt I c) c))
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
dg : mfderiv I I g (↑(extChartAt I c) c) = 0
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg' : HasFDerivAt g 0 (↑(extChartAt I c) c)
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
dg : mfderiv I I g (↑(extChartAt I c) c) = 0
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg' : HasMFDerivAt I I g (↑(extChartAt I c) c) (mfderiv I I g (↑(extChartAt I c) c))
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | replace dg := dg'.hasDerivAt | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
dg : mfderiv I I g (↑(extChartAt I c) c) = 0
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg' : HasFDerivAt g 0 (↑(extChartAt I c) c)
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg' : HasFDerivAt g 0 (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
dg : mfderiv I I g (↑(extChartAt I c) c) = 0
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg' : HasFDerivAt g 0 (↑(extChartAt I c) c)
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | clear dg' | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg' : HasFDerivAt g 0 (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg' : HasFDerivAt g 0 (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rcases not_local_inj_of_deriv_zero fa.2 dg with ⟨h, ha, h0, e⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | refine ⟨fun z ↦ (extChartAt I c).symm (h (extChartAt I c z)), ?_, ?_, ?_⟩ | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z | case intro.intro.intro.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ HolomorphicAt I I (fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) c
case intro.intro.intro.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ (fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) c = c
case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ ∀ᶠ (z : S) in 𝓝[≠] c,
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) z ≠ z ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ ∃ g, HolomorphicAt I I g c ∧ g c = c ∧ ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | have fd : MDifferentiableAt I I f ((extChartAt I c).symm (extChartAt I c c)) := by
rw [PartialEquiv.left_inv]; exact fa.mdifferentiableAt; apply mem_extChartAt_source | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ mfderiv I I g (↑(extChartAt I c) c) = 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ mfderiv I I g (↑(extChartAt I c) c) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ mfderiv I I g (↑(extChartAt I c) c) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [← hg, ←Function.comp_def, mfderiv_comp _ (HolomorphicAt.extChartAt _).mdifferentiableAt _,
←Function.comp_def, mfderiv_comp _ fd (HolomorphicAt.extChartAt_symm _).mdifferentiableAt,
PartialEquiv.left_inv, df, ContinuousLinearMap.zero_comp, ContinuousLinearMap.comp_zero] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ mfderiv I I g (↑(extChartAt I c) c) = 0 | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ c ∈ (extChartAt I c).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ ↑(extChartAt I c) c ∈ (extChartAt I c).target
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ f (↑(extChartAt I c).symm (↑(extChartAt I c) c)) ∈ (extChartAt I (f c)).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ mfderiv I I g (↑(extChartAt I c) c) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply mem_extChartAt_source | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ c ∈ (extChartAt I c).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ ↑(extChartAt I c) c ∈ (extChartAt I c).target
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ f (↑(extChartAt I c).symm (↑(extChartAt I c) c)) ∈ (extChartAt I (f c)).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ ↑(extChartAt I c) c ∈ (extChartAt I c).target
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ f (↑(extChartAt I c).symm (↑(extChartAt I c) c)) ∈ (extChartAt I (f c)).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ c ∈ (extChartAt I c).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ ↑(extChartAt I c) c ∈ (extChartAt I c).target
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ f (↑(extChartAt I c).symm (↑(extChartAt I c) c)) ∈ (extChartAt I (f c)).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply mem_extChartAt_target | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ ↑(extChartAt I c) c ∈ (extChartAt I c).target
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ f (↑(extChartAt I c).symm (↑(extChartAt I c) c)) ∈ (extChartAt I (f c)).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ f (↑(extChartAt I c).symm (↑(extChartAt I c) c)) ∈ (extChartAt I (f c)).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ ↑(extChartAt I c) c ∈ (extChartAt I c).target
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ f (↑(extChartAt I c).symm (↑(extChartAt I c) c)) ∈ (extChartAt I (f c)).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [PartialEquiv.left_inv] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ f (↑(extChartAt I c).symm (↑(extChartAt I c) c)) ∈ (extChartAt I (f c)).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ f c ∈ (extChartAt I (f c)).source
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ c ∈ (extChartAt I c).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ f (↑(extChartAt I c).symm (↑(extChartAt I c) c)) ∈ (extChartAt I (f c)).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply mem_extChartAt_source | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ f c ∈ (extChartAt I (f c)).source
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ c ∈ (extChartAt I c).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c) | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ c ∈ (extChartAt I c).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ f c ∈ (extChartAt I (f c)).source
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ c ∈ (extChartAt I c).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply mem_extChartAt_source | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ c ∈ (extChartAt I c).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ c ∈ (extChartAt I c).source
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | exact MDifferentiableAt.comp _ fd
(HolomorphicAt.extChartAt_symm (mem_extChartAt_target _ _)).mdifferentiableAt | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fd : MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
⊢ MDifferentiableAt I I (fun z => f (↑(extChartAt I c).symm z)) (↑(extChartAt I c) c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [PartialEquiv.left_inv] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c)) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ MDifferentiableAt I I f c
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ c ∈ (extChartAt I c).source | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ MDifferentiableAt I I f (↑(extChartAt I c).symm (↑(extChartAt I c) c))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | exact fa.mdifferentiableAt | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ MDifferentiableAt I I f c
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ c ∈ (extChartAt I c).source | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ c ∈ (extChartAt I c).source | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ MDifferentiableAt I I f c
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ c ∈ (extChartAt I c).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply mem_extChartAt_source | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ c ∈ (extChartAt I c).source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
fa : HolomorphicAt I I f c
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
⊢ c ∈ (extChartAt I c).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply (HolomorphicAt.extChartAt_symm (mem_extChartAt_target I c)).comp_of_eq | case intro.intro.intro.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ HolomorphicAt I I (fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) c | case intro.intro.intro.refine_1.gh
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ HolomorphicAt I I (fun x => h (↑(extChartAt I c) x)) c
case intro.intro.intro.refine_1.e
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ HolomorphicAt I I (fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply (ha.holomorphicAt I I).comp_of_eq
(HolomorphicAt.extChartAt (mem_extChartAt_source I c)) rfl | case intro.intro.intro.refine_1.gh
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ HolomorphicAt I I (fun x => h (↑(extChartAt I c) x)) c
case intro.intro.intro.refine_1.e
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c | case intro.intro.intro.refine_1.e
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_1.gh
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ HolomorphicAt I I (fun x => h (↑(extChartAt I c) x)) c
case intro.intro.intro.refine_1.e
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | exact h0 | case intro.intro.intro.refine_1.e
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_1.e
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | simp only [h0, PartialEquiv.left_inv _ (mem_extChartAt_source I c)] | case intro.intro.intro.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ (fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) c = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ (fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) c = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [eventually_nhdsWithin_iff] at e ⊢ | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ ∀ᶠ (z : S) in 𝓝[≠] c,
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) z ≠ z ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) z) = f z | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ∀ᶠ (x : S) in 𝓝 c,
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (z : ℂ) in 𝓝[≠] ↑(extChartAt I c) c, h z ≠ z ∧ g (h z) = g z
⊢ ∀ᶠ (z : S) in 𝓝[≠] c,
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) z ≠ z ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply ((continuousAt_extChartAt I c).eventually e).mp | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ∀ᶠ (x : S) in 𝓝 c,
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ∀ᶠ (x : S) in 𝓝 c,
(↑(extChartAt I c) x ∈ {↑(extChartAt I c) c}ᶜ →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ∀ᶠ (x : S) in 𝓝 c,
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply ((isOpen_extChartAt_source I c).eventually_mem (mem_extChartAt_source I c)).mp | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ∀ᶠ (x : S) in 𝓝 c,
(↑(extChartAt I c) x ∈ {↑(extChartAt I c) c}ᶜ →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ∀ᶠ (x : S) in 𝓝 c,
x ∈ (extChartAt I c).source →
(↑(extChartAt I c) x ∈ {↑(extChartAt I c) c}ᶜ →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ∀ᶠ (x : S) in 𝓝 c,
(↑(extChartAt I c) x ∈ {↑(extChartAt I c) c}ᶜ →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | have m2 : ∀ᶠ z in 𝓝 c, f z ∈ (extChartAt I (f c)).source :=
fa.1.eventually_mem (extChartAt_source_mem_nhds I _) | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
⊢ ∀ᶠ (x : S) in 𝓝 c,
x ∈ (extChartAt I c).source →
(↑(extChartAt I c) x ∈ {↑(extChartAt I c) c}ᶜ →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ∀ᶠ (x : S) in 𝓝 c,
x ∈ (extChartAt I c).source →
(↑(extChartAt I c) x ∈ {↑(extChartAt I c) c}ᶜ →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
⊢ ∀ᶠ (x : S) in 𝓝 c,
x ∈ (extChartAt I c).source →
(↑(extChartAt I c) x ∈ {↑(extChartAt I c) c}ᶜ →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | refine m1.mp (m2.mp (m3.mp (eventually_of_forall ?_))) | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3 : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
⊢ ∀ᶠ (x : S) in 𝓝 c,
x ∈ (extChartAt I c).source →
(↑(extChartAt I c) x ∈ {↑(extChartAt I c) c}ᶜ →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3 : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
⊢ ∀ (x : S),
f (↑(extChartAt I c).symm (h (↑(extChartAt I c) x))) ∈ (extChartAt I (f c)).source →
f x ∈ (extChartAt I (f c)).source →
h (↑(extChartAt I c) x) ∈ (extChartAt I c).target →
x ∈ (extChartAt I c).source →
(↑(extChartAt I c) x ∈ {↑(extChartAt I c) c}ᶜ →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3 : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
⊢ ∀ᶠ (x : S) in 𝓝 c,
x ∈ (extChartAt I c).source →
(↑(extChartAt I c) x ∈ {↑(extChartAt I c) c}ᶜ →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | simp only [mem_compl_singleton_iff] | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3 : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
⊢ ∀ (x : S),
f (↑(extChartAt I c).symm (h (↑(extChartAt I c) x))) ∈ (extChartAt I (f c)).source →
f x ∈ (extChartAt I (f c)).source →
h (↑(extChartAt I c) x) ∈ (extChartAt I c).target →
x ∈ (extChartAt I c).source →
(↑(extChartAt I c) x ∈ {↑(extChartAt I c) c}ᶜ →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3 : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
⊢ ∀ (x : S),
f (↑(extChartAt I c).symm (h (↑(extChartAt I c) x))) ∈ (extChartAt I (f c)).source →
f x ∈ (extChartAt I (f c)).source →
h (↑(extChartAt I c) x) ∈ (extChartAt I c).target →
x ∈ (extChartAt I c).source →
(↑(extChartAt I c) x ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ≠ c →
↑(extChartAt I c).symm (h (↑(extChartAt I c) x)) ≠ x ∧
f (↑(extChartAt I c).symm (h (↑(extChartAt I c) x))) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3 : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
⊢ ∀ (x : S),
f (↑(extChartAt I c).symm (h (↑(extChartAt I c) x))) ∈ (extChartAt I (f c)).source →
f x ∈ (extChartAt I (f c)).source →
h (↑(extChartAt I c) x) ∈ (extChartAt I c).target →
x ∈ (extChartAt I c).source →
(↑(extChartAt I c) x ∈ {↑(extChartAt I c) c}ᶜ →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ∈ {c}ᶜ →
(fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x ≠ x ∧
f ((fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) x) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | intro z m3 m2 m1 m0 even zc | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3 : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
⊢ ∀ (x : S),
f (↑(extChartAt I c).symm (h (↑(extChartAt I c) x))) ∈ (extChartAt I (f c)).source →
f x ∈ (extChartAt I (f c)).source →
h (↑(extChartAt I c) x) ∈ (extChartAt I c).target →
x ∈ (extChartAt I c).source →
(↑(extChartAt I c) x ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ≠ c →
↑(extChartAt I c).symm (h (↑(extChartAt I c) x)) ≠ x ∧
f (↑(extChartAt I c).symm (h (↑(extChartAt I c) x))) = f x | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) z)) ≠ z ∧ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3 : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
⊢ ∀ (x : S),
f (↑(extChartAt I c).symm (h (↑(extChartAt I c) x))) ∈ (extChartAt I (f c)).source →
f x ∈ (extChartAt I (f c)).source →
h (↑(extChartAt I c) x) ∈ (extChartAt I c).target →
x ∈ (extChartAt I c).source →
(↑(extChartAt I c) x ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) x) ≠ ↑(extChartAt I c) x ∧ g (h (↑(extChartAt I c) x)) = g (↑(extChartAt I c) x)) →
x ≠ c →
↑(extChartAt I c).symm (h (↑(extChartAt I c) x)) ≠ x ∧
f (↑(extChartAt I c).symm (h (↑(extChartAt I c) x))) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rcases even ((PartialEquiv.injOn _).ne m0 (mem_extChartAt_source I c) zc) with ⟨hz, gh⟩ | case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) z)) ≠ z ∧ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z | case intro.intro.intro.refine_3.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) z)) ≠ z ∧ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) z)) ≠ z ∧ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | constructor | case intro.intro.intro.refine_3.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) z)) ≠ z ∧ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z | case intro.intro.intro.refine_3.intro.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) z)) ≠ z
case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) z)) ≠ z ∧ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | refine ContinuousAt.eventually_mem ?_ (extChartAt_target_mem_nhds' I ?_) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target | case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ContinuousAt (fun z => h (↑(extChartAt I c) z)) c
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ h (↑(extChartAt I c) c) ∈ (extChartAt I c).target | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | exact ha.continuousAt.comp_of_eq (continuousAt_extChartAt I c) rfl | case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ContinuousAt (fun z => h (↑(extChartAt I c) z)) c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ContinuousAt (fun z => h (↑(extChartAt I c) z)) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [h0] | case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ h (↑(extChartAt I c) c) ∈ (extChartAt I c).target | case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ↑(extChartAt I c) c ∈ (extChartAt I c).target | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ h (↑(extChartAt I c) c) ∈ (extChartAt I c).target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | exact mem_extChartAt_target I c | case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ↑(extChartAt I c) c ∈ (extChartAt I c).target | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
⊢ ↑(extChartAt I c) c ∈ (extChartAt I c).target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | refine ContinuousAt.eventually_mem ?_ (extChartAt_source_mem_nhds' I ?_) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source | case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ContinuousAt (fun z => f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z)))) c
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) c))) ∈ (extChartAt I (f c)).source | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply fa.1.comp_of_eq | case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ContinuousAt (fun z => f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z)))) c | case refine_1.hf
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ContinuousAt (fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ContinuousAt (fun z => f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z)))) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply (continuousAt_extChartAt_symm I _).comp_of_eq | case refine_1.hf
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ContinuousAt (fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c | case refine_1.hf.hf
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ContinuousAt (fun z => h (↑(extChartAt I c) z)) c
case refine_1.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ContinuousAt (fun z => ↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply ha.continuousAt.comp_of_eq | case refine_1.hf.hf
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ContinuousAt (fun z => h (↑(extChartAt I c) z)) c
case refine_1.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c | case refine_1.hf.hf.hf
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ContinuousAt (fun z => ↑(extChartAt I c) z) c
case refine_1.hf.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c) c = ↑(extChartAt I c) c
case refine_1.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf.hf
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ContinuousAt (fun z => h (↑(extChartAt I c) z)) c
case refine_1.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | exact continuousAt_extChartAt I _ | case refine_1.hf.hf.hf
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ContinuousAt (fun z => ↑(extChartAt I c) z) c
case refine_1.hf.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c) c = ↑(extChartAt I c) c
case refine_1.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c | case refine_1.hf.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c) c = ↑(extChartAt I c) c
case refine_1.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf.hf.hf
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ContinuousAt (fun z => ↑(extChartAt I c) z) c
case refine_1.hf.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c) c = ↑(extChartAt I c) c
case refine_1.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rfl | case refine_1.hf.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c) c = ↑(extChartAt I c) c
case refine_1.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c | case refine_1.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c) c = ↑(extChartAt I c) c
case refine_1.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | exact h0 | case refine_1.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c | case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [h0, PartialEquiv.left_inv _ (mem_extChartAt_source I _)] | case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) c)) = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [h0, PartialEquiv.left_inv _ (mem_extChartAt_source I _)] | case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) c))) ∈ (extChartAt I (f c)).source | case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ f c ∈ (extChartAt I (f c)).source | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) c))) ∈ (extChartAt I (f c)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | apply mem_extChartAt_source | case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ f c ∈ (extChartAt I (f c)).source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1 : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2 : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
⊢ f c ∈ (extChartAt I (f c)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | nth_rw 2 [← PartialEquiv.left_inv _ m0] | case intro.intro.intro.refine_3.intro.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) z)) ≠ z | case intro.intro.intro.refine_3.intro.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) z)) ≠ ↑(extChartAt I c).symm (↑(extChartAt I c) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) z)) ≠ z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [(PartialEquiv.injOn _).ne_iff] | case intro.intro.intro.refine_3.intro.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) z)) ≠ ↑(extChartAt I c).symm (↑(extChartAt I c) z) | case intro.intro.intro.refine_3.intro.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
case intro.intro.intro.refine_3.intro.left.hx
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ h (↑(extChartAt I c) z) ∈ (extChartAt I c).symm.source
case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).symm.source | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c).symm (h (↑(extChartAt I c) z)) ≠ ↑(extChartAt I c).symm (↑(extChartAt I c) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | exact hz | case intro.intro.intro.refine_3.intro.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
case intro.intro.intro.refine_3.intro.left.hx
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ h (↑(extChartAt I c) z) ∈ (extChartAt I c).symm.source
case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).symm.source | case intro.intro.intro.refine_3.intro.left.hx
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ h (↑(extChartAt I c) z) ∈ (extChartAt I c).symm.source
case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).symm.source | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
case intro.intro.intro.refine_3.intro.left.hx
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ h (↑(extChartAt I c) z) ∈ (extChartAt I c).symm.source
case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).symm.source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [PartialEquiv.symm_source] | case intro.intro.intro.refine_3.intro.left.hx
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ h (↑(extChartAt I c) z) ∈ (extChartAt I c).symm.source
case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).symm.source | case intro.intro.intro.refine_3.intro.left.hx
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).symm.source | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.left.hx
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ h (↑(extChartAt I c) z) ∈ (extChartAt I c).symm.source
case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).symm.source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | exact m1 | case intro.intro.intro.refine_3.intro.left.hx
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).symm.source | case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).symm.source | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.left.hx
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).symm.source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [PartialEquiv.symm_source] | case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).symm.source | case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).target | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).symm.source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | exact PartialEquiv.map_source _ m0 | case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).target | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.left.hy
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ ↑(extChartAt I c) z ∈ (extChartAt I c).target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | simp only [← hg] at gh | case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z | case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh :
↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z)))) =
↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm (↑(extChartAt I c) z)))
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [PartialEquiv.left_inv _ m0] at gh | case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh :
↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z)))) =
↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm (↑(extChartAt I c) z)))
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z | case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z)))) = ↑(extChartAt I (f c)) (f z)
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh :
↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z)))) =
↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm (↑(extChartAt I c) z)))
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | rw [(PartialEquiv.injOn _).eq_iff m3 m2] at gh | case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z)))) = ↑(extChartAt I (f c)) (f z)
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z | case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z)))) = ↑(extChartAt I (f c)) (f z)
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_mfderiv_zero | [172, 1] | [225, 63] | exact gh | case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
c : S
df : mfderiv I I f c = 0
g : ℂ → ℂ
hg : (fun z => ↑(extChartAt I (f c)) (f (↑(extChartAt I c).symm z))) = g
fa : ContinuousAt f c ∧ AnalyticAt ℂ g (↑(extChartAt I c) c)
dg : HasDerivAt g (0 1) (↑(extChartAt I c) c)
h : ℂ → ℂ
ha : AnalyticAt ℂ h (↑(extChartAt I c) c)
h0 : h (↑(extChartAt I c) c) = ↑(extChartAt I c) c
e : ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I c) c), x ∈ {↑(extChartAt I c) c}ᶜ → h x ≠ x ∧ g (h x) = g x
m1✝ : ∀ᶠ (z : S) in 𝓝 c, h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m2✝ : ∀ᶠ (z : S) in 𝓝 c, f z ∈ (extChartAt I (f c)).source
m3✝ : ∀ᶠ (z : S) in 𝓝 c, f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
z : S
m3 : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) ∈ (extChartAt I (f c)).source
m2 : f z ∈ (extChartAt I (f c)).source
m1 : h (↑(extChartAt I c) z) ∈ (extChartAt I c).target
m0 : z ∈ (extChartAt I c).source
even :
↑(extChartAt I c) z ≠ ↑(extChartAt I c) c →
h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z ∧ g (h (↑(extChartAt I c) z)) = g (↑(extChartAt I c) z)
zc : z ≠ c
hz : h (↑(extChartAt I c) z) ≠ ↑(extChartAt I c) z
gh : f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z
⊢ f (↑(extChartAt I c).symm (h (↑(extChartAt I c) z))) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | Set.InjOn.mfderiv_ne_zero | [228, 1] | [237, 29] | contrapose inj | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
inj : InjOn f s
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
⊢ mfderiv I I f c ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : ¬mfderiv I I f c ≠ 0
⊢ ¬InjOn f s | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
inj : InjOn f s
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
⊢ mfderiv I I f c ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | Set.InjOn.mfderiv_ne_zero | [228, 1] | [237, 29] | simp only [not_not, InjOn, not_forall] at inj ⊢ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : ¬mfderiv I I f c ≠ 0
⊢ ¬InjOn f s | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : ¬mfderiv I I f c ≠ 0
⊢ ¬InjOn f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | Set.InjOn.mfderiv_ne_zero | [228, 1] | [237, 29] | rcases not_local_inj_of_mfderiv_zero fa inj with ⟨g, ga, gc, fg⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1 | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
fg : ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | Set.InjOn.mfderiv_ne_zero | [228, 1] | [237, 29] | have gm : ∀ᶠ z in 𝓝 c, g z ∈ s :=
ga.continuousAt.eventually_mem (so.mem_nhds (by simp only [gc, m])) | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
fg : ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1 | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
fg : ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
gm : ∀ᶠ (z : S) in 𝓝 c, g z ∈ s
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
fg : ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | Set.InjOn.mfderiv_ne_zero | [228, 1] | [237, 29] | replace fg := fg.and (((so.eventually_mem m).and gm).filter_mono nhdsWithin_le_nhds) | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
fg : ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
gm : ∀ᶠ (z : S) in 𝓝 c, g z ∈ s
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1 | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
gm : ∀ᶠ (z : S) in 𝓝 c, g z ∈ s
fg : ∀ᶠ (x : S) in 𝓝[≠] c, (g x ≠ x ∧ f (g x) = f x) ∧ x ∈ s ∧ g x ∈ s
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
fg : ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
gm : ∀ᶠ (z : S) in 𝓝 c, g z ∈ s
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | Set.InjOn.mfderiv_ne_zero | [228, 1] | [237, 29] | rcases @Filter.Eventually.exists _ _ _ (AnalyticManifold.punctured_nhds_neBot I c) fg
with ⟨z, ⟨gz, fg⟩, zs, gs⟩ | case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
gm : ∀ᶠ (z : S) in 𝓝 c, g z ∈ s
fg : ∀ᶠ (x : S) in 𝓝[≠] c, (g x ≠ x ∧ f (g x) = f x) ∧ x ∈ s ∧ g x ∈ s
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1 | case intro.intro.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
gm : ∀ᶠ (z : S) in 𝓝 c, g z ∈ s
fg✝ : ∀ᶠ (x : S) in 𝓝[≠] c, (g x ≠ x ∧ f (g x) = f x) ∧ x ∈ s ∧ g x ∈ s
z : S
gz : g z ≠ z
fg : f (g z) = f z
zs : z ∈ s
gs : g z ∈ s
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
gm : ∀ᶠ (z : S) in 𝓝 c, g z ∈ s
fg : ∀ᶠ (x : S) in 𝓝[≠] c, (g x ≠ x ∧ f (g x) = f x) ∧ x ∈ s ∧ g x ∈ s
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | Set.InjOn.mfderiv_ne_zero | [228, 1] | [237, 29] | use g z, gs, z, zs, fg, gz | case intro.intro.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
gm : ∀ᶠ (z : S) in 𝓝 c, g z ∈ s
fg✝ : ∀ᶠ (x : S) in 𝓝[≠] c, (g x ≠ x ∧ f (g x) = f x) ∧ x ∈ s ∧ g x ∈ s
z : S
gz : g z ≠ z
fg : f (g z) = f z
zs : z ∈ s
gs : g z ∈ s
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
gm : ∀ᶠ (z : S) in 𝓝 c, g z ∈ s
fg✝ : ∀ᶠ (x : S) in 𝓝[≠] c, (g x ≠ x ∧ f (g x) = f x) ∧ x ∈ s ∧ g x ∈ s
z : S
gz : g z ≠ z
fg : f (g z) = f z
zs : z ∈ s
gs : g z ∈ s
⊢ ∃ x, ∃ (_ : x ∈ s), ∃ x_1, ∃ (_ : x_1 ∈ s) (_ : f x = f x_1), ¬x = x_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | Set.InjOn.mfderiv_ne_zero | [228, 1] | [237, 29] | simp only [gc, m] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
fg : ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
⊢ g c ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : S → T
s : Set S
so : IsOpen s
c : S
m : c ∈ s
fa : HolomorphicAt I I f c
inj : mfderiv I I f c = 0
g : S → S
ga : HolomorphicAt I I g c
gc : g c = c
fg : ∀ᶠ (z : S) in 𝓝[≠] c, g z ≠ z ∧ f (g z) = f z
⊢ g c ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | constructor | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
⊢ DifferentiableOn ℂ f s ↔ AnalyticOn ℂ f s | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
⊢ DifferentiableOn ℂ f s → AnalyticOn ℂ f s
case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
⊢ AnalyticOn ℂ f s → DifferentiableOn ℂ f s | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
⊢ DifferentiableOn ℂ f s ↔ AnalyticOn ℂ f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | intro d | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
⊢ DifferentiableOn ℂ f s → AnalyticOn ℂ f s | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
⊢ AnalyticOn ℂ f s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
⊢ DifferentiableOn ℂ f s → AnalyticOn ℂ f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | apply osgood o d.continuousOn | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
⊢ AnalyticOn ℂ f s | case mp.fa0
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
⊢ ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
case mp.fa1
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
⊢ ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (z0, z1)) z1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
⊢ AnalyticOn ℂ f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | intro z0 z1 zs | case mp.fa0
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
⊢ ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0 | case mp.fa0
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa0
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
⊢ ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | rcases Metric.isOpen_iff.mp o (z0, z1) zs with ⟨r, rp, rs⟩ | case mp.fa0
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0 | case mp.fa0.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa0
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | have d0 : DifferentiableOn ℂ (fun z0 ↦ f (z0, z1)) (ball z0 r) := by
apply DifferentiableOn.comp d
exact DifferentiableOn.prod differentiableOn_id (differentiableOn_const _)
intro z0 z0s; apply rs; simp at z0s ⊢; assumption | case mp.fa0.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0 | case mp.fa0.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
d0 : DifferentiableOn ℂ (fun z0 => f (z0, z1)) (ball z0 r)
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa0.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | exact (analyticOn_iff_differentiableOn isOpen_ball).mpr d0 z0 (Metric.mem_ball_self rp) | case mp.fa0.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
d0 : DifferentiableOn ℂ (fun z0 => f (z0, z1)) (ball z0 r)
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa0.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
d0 : DifferentiableOn ℂ (fun z0 => f (z0, z1)) (ball z0 r)
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | apply DifferentiableOn.comp d | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ DifferentiableOn ℂ (fun z0 => f (z0, z1)) (ball z0 r) | case hf
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ DifferentiableOn ℂ (fun z0 => (z0, z1)) (ball z0 r)
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ Set.MapsTo (fun z0 => (z0, z1)) (ball z0 r) s | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ DifferentiableOn ℂ (fun z0 => f (z0, z1)) (ball z0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | exact DifferentiableOn.prod differentiableOn_id (differentiableOn_const _) | case hf
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ DifferentiableOn ℂ (fun z0 => (z0, z1)) (ball z0 r)
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ Set.MapsTo (fun z0 => (z0, z1)) (ball z0 r) s | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ Set.MapsTo (fun z0 => (z0, z1)) (ball z0 r) s | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ DifferentiableOn ℂ (fun z0 => (z0, z1)) (ball z0 r)
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ Set.MapsTo (fun z0 => (z0, z1)) (ball z0 r) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | intro z0 z0s | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ Set.MapsTo (fun z0 => (z0, z1)) (ball z0 r) s | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0✝ z1 : ℂ
zs : (z0✝, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0✝, z1) r ⊆ s
z0 : ℂ
z0s : z0 ∈ ball z0✝ r
⊢ (fun z0 => (z0, z1)) z0 ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ Set.MapsTo (fun z0 => (z0, z1)) (ball z0 r) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | apply rs | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0✝ z1 : ℂ
zs : (z0✝, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0✝, z1) r ⊆ s
z0 : ℂ
z0s : z0 ∈ ball z0✝ r
⊢ (fun z0 => (z0, z1)) z0 ∈ s | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0✝ z1 : ℂ
zs : (z0✝, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0✝, z1) r ⊆ s
z0 : ℂ
z0s : z0 ∈ ball z0✝ r
⊢ (fun z0 => (z0, z1)) z0 ∈ ball (z0✝, z1) r | Please generate a tactic in lean4 to solve the state.
STATE:
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0✝ z1 : ℂ
zs : (z0✝, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0✝, z1) r ⊆ s
z0 : ℂ
z0s : z0 ∈ ball z0✝ r
⊢ (fun z0 => (z0, z1)) z0 ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | simp at z0s ⊢ | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0✝ z1 : ℂ
zs : (z0✝, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0✝, z1) r ⊆ s
z0 : ℂ
z0s : z0 ∈ ball z0✝ r
⊢ (fun z0 => (z0, z1)) z0 ∈ ball (z0✝, z1) r | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0✝ z1 : ℂ
zs : (z0✝, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0✝, z1) r ⊆ s
z0 : ℂ
z0s : dist z0 z0✝ < r
⊢ dist z0 z0✝ < r | Please generate a tactic in lean4 to solve the state.
STATE:
case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0✝ z1 : ℂ
zs : (z0✝, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0✝, z1) r ⊆ s
z0 : ℂ
z0s : z0 ∈ ball z0✝ r
⊢ (fun z0 => (z0, z1)) z0 ∈ ball (z0✝, z1) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | assumption | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0✝ z1 : ℂ
zs : (z0✝, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0✝, z1) r ⊆ s
z0 : ℂ
z0s : dist z0 z0✝ < r
⊢ dist z0 z0✝ < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0✝ z1 : ℂ
zs : (z0✝, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0✝, z1) r ⊆ s
z0 : ℂ
z0s : dist z0 z0✝ < r
⊢ dist z0 z0✝ < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | intro z0 z1 zs | case mp.fa1
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
⊢ ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (z0, z1)) z1 | case mp.fa1
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa1
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
⊢ ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | rcases Metric.isOpen_iff.mp o (z0, z1) zs with ⟨r, rp, rs⟩ | case mp.fa1
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1 | case mp.fa1.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa1
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | have d1 : DifferentiableOn ℂ (fun z1 ↦ f (z0, z1)) (ball z1 r) := by
apply DifferentiableOn.comp d
exact DifferentiableOn.prod (differentiableOn_const _) differentiableOn_id
intro z1 z1s; apply rs; simp at z1s ⊢; assumption | case mp.fa1.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1 | case mp.fa1.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
d1 : DifferentiableOn ℂ (fun z1 => f (z0, z1)) (ball z1 r)
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa1.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | exact (analyticOn_iff_differentiableOn isOpen_ball).mpr d1 z1 (Metric.mem_ball_self rp) | case mp.fa1.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
d1 : DifferentiableOn ℂ (fun z1 => f (z0, z1)) (ball z1 r)
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa1.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
d1 : DifferentiableOn ℂ (fun z1 => f (z0, z1)) (ball z1 r)
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | apply DifferentiableOn.comp d | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ DifferentiableOn ℂ (fun z1 => f (z0, z1)) (ball z1 r) | case hf
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ DifferentiableOn ℂ (fun z1 => (z0, z1)) (ball z1 r)
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ Set.MapsTo (fun z1 => (z0, z1)) (ball z1 r) s | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ DifferentiableOn ℂ (fun z1 => f (z0, z1)) (ball z1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | exact DifferentiableOn.prod (differentiableOn_const _) differentiableOn_id | case hf
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ DifferentiableOn ℂ (fun z1 => (z0, z1)) (ball z1 r)
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ Set.MapsTo (fun z1 => (z0, z1)) (ball z1 r) s | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ Set.MapsTo (fun z1 => (z0, z1)) (ball z1 r) s | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ DifferentiableOn ℂ (fun z1 => (z0, z1)) (ball z1 r)
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ Set.MapsTo (fun z1 => (z0, z1)) (ball z1 r) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | intro z1 z1s | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ Set.MapsTo (fun z1 => (z0, z1)) (ball z1 r) s | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1✝ : ℂ
zs : (z0, z1✝) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1✝) r ⊆ s
z1 : ℂ
z1s : z1 ∈ ball z1✝ r
⊢ (fun z1 => (z0, z1)) z1 ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1 : ℂ
zs : (z0, z1) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1) r ⊆ s
⊢ Set.MapsTo (fun z1 => (z0, z1)) (ball z1 r) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | apply rs | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1✝ : ℂ
zs : (z0, z1✝) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1✝) r ⊆ s
z1 : ℂ
z1s : z1 ∈ ball z1✝ r
⊢ (fun z1 => (z0, z1)) z1 ∈ s | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1✝ : ℂ
zs : (z0, z1✝) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1✝) r ⊆ s
z1 : ℂ
z1s : z1 ∈ ball z1✝ r
⊢ (fun z1 => (z0, z1)) z1 ∈ ball (z0, z1✝) r | Please generate a tactic in lean4 to solve the state.
STATE:
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1✝ : ℂ
zs : (z0, z1✝) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1✝) r ⊆ s
z1 : ℂ
z1s : z1 ∈ ball z1✝ r
⊢ (fun z1 => (z0, z1)) z1 ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | simp at z1s ⊢ | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1✝ : ℂ
zs : (z0, z1✝) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1✝) r ⊆ s
z1 : ℂ
z1s : z1 ∈ ball z1✝ r
⊢ (fun z1 => (z0, z1)) z1 ∈ ball (z0, z1✝) r | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1✝ : ℂ
zs : (z0, z1✝) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1✝) r ⊆ s
z1 : ℂ
z1s : dist z1 z1✝ < r
⊢ dist z1 z1✝ < r | Please generate a tactic in lean4 to solve the state.
STATE:
case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1✝ : ℂ
zs : (z0, z1✝) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1✝) r ⊆ s
z1 : ℂ
z1s : z1 ∈ ball z1✝ r
⊢ (fun z1 => (z0, z1)) z1 ∈ ball (z0, z1✝) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | assumption | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1✝ : ℂ
zs : (z0, z1✝) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1✝) r ⊆ s
z1 : ℂ
z1s : dist z1 z1✝ < r
⊢ dist z1 z1✝ < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : DifferentiableOn ℂ f s
z0 z1✝ : ℂ
zs : (z0, z1✝) ∈ s
r : ℝ
rp : r > 0
rs : ball (z0, z1✝) r ⊆ s
z1 : ℂ
z1s : dist z1 z1✝ < r
⊢ dist z1 z1✝ < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | exact fun a ↦ a.differentiableOn | case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
⊢ AnalyticOn ℂ f s → DifferentiableOn ℂ f s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
⊢ AnalyticOn ℂ f s → DifferentiableOn ℂ f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | constructor | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
⊢ ContDiffAt ℂ n f x ↔ AnalyticAt ℂ f x | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
⊢ ContDiffAt ℂ n f x → AnalyticAt ℂ f x
case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
⊢ AnalyticAt ℂ f x → ContDiffAt ℂ n f x | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
⊢ ContDiffAt ℂ n f x ↔ AnalyticAt ℂ f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | intro d | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
⊢ ContDiffAt ℂ n f x → AnalyticAt ℂ f x | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d : ContDiffAt ℂ n f x
⊢ AnalyticAt ℂ f x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
⊢ ContDiffAt ℂ n f x → AnalyticAt ℂ f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | rcases d.contDiffOn n1 with ⟨u, un, d⟩ | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d : ContDiffAt ℂ n f x
⊢ AnalyticAt ℂ f x | case mp.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d✝ : ContDiffAt ℂ n f x
u : Set (ℂ × ℂ)
un : u ∈ 𝓝 x
d : ContDiffOn ℂ (↑One.one) f u
⊢ AnalyticAt ℂ f x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d : ContDiffAt ℂ n f x
⊢ AnalyticAt ℂ f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | rcases mem_nhds_iff.mp un with ⟨v, uv, vo, vx⟩ | case mp.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d✝ : ContDiffAt ℂ n f x
u : Set (ℂ × ℂ)
un : u ∈ 𝓝 x
d : ContDiffOn ℂ (↑One.one) f u
⊢ AnalyticAt ℂ f x | case mp.intro.intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d✝ : ContDiffAt ℂ n f x
u : Set (ℂ × ℂ)
un : u ∈ 𝓝 x
d : ContDiffOn ℂ (↑One.one) f u
v : Set (ℂ × ℂ)
uv : v ⊆ u
vo : IsOpen v
vx : x ∈ v
⊢ AnalyticAt ℂ f x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d✝ : ContDiffAt ℂ n f x
u : Set (ℂ × ℂ)
un : u ∈ 𝓝 x
d : ContDiffOn ℂ (↑One.one) f u
⊢ AnalyticAt ℂ f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | refine (differentiable_iff_analytic2 vo).mp ?_ _ vx | case mp.intro.intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d✝ : ContDiffAt ℂ n f x
u : Set (ℂ × ℂ)
un : u ∈ 𝓝 x
d : ContDiffOn ℂ (↑One.one) f u
v : Set (ℂ × ℂ)
uv : v ⊆ u
vo : IsOpen v
vx : x ∈ v
⊢ AnalyticAt ℂ f x | case mp.intro.intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d✝ : ContDiffAt ℂ n f x
u : Set (ℂ × ℂ)
un : u ∈ 𝓝 x
d : ContDiffOn ℂ (↑One.one) f u
v : Set (ℂ × ℂ)
uv : v ⊆ u
vo : IsOpen v
vx : x ∈ v
⊢ DifferentiableOn ℂ f v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d✝ : ContDiffAt ℂ n f x
u : Set (ℂ × ℂ)
un : u ∈ 𝓝 x
d : ContDiffOn ℂ (↑One.one) f u
v : Set (ℂ × ℂ)
uv : v ⊆ u
vo : IsOpen v
vx : x ∈ v
⊢ AnalyticAt ℂ f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | exact (d.mono uv).differentiableOn (by norm_num) | case mp.intro.intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d✝ : ContDiffAt ℂ n f x
u : Set (ℂ × ℂ)
un : u ∈ 𝓝 x
d : ContDiffOn ℂ (↑One.one) f u
v : Set (ℂ × ℂ)
uv : v ⊆ u
vo : IsOpen v
vx : x ∈ v
⊢ DifferentiableOn ℂ f v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d✝ : ContDiffAt ℂ n f x
u : Set (ℂ × ℂ)
un : u ∈ 𝓝 x
d : ContDiffOn ℂ (↑One.one) f u
v : Set (ℂ × ℂ)
uv : v ⊆ u
vo : IsOpen v
vx : x ∈ v
⊢ DifferentiableOn ℂ f v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | norm_num | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d✝ : ContDiffAt ℂ n f x
u : Set (ℂ × ℂ)
un : u ∈ 𝓝 x
d : ContDiffOn ℂ (↑One.one) f u
v : Set (ℂ × ℂ)
uv : v ⊆ u
vo : IsOpen v
vx : x ∈ v
⊢ 1 ≤ ↑One.one | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d✝ : ContDiffAt ℂ n f x
u : Set (ℂ × ℂ)
un : u ∈ 𝓝 x
d : ContDiffOn ℂ (↑One.one) f u
v : Set (ℂ × ℂ)
uv : v ⊆ u
vo : IsOpen v
vx : x ∈ v
⊢ 1 ≤ ↑One.one
TACTIC:
|
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