url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_ne_zero_eventually | [355, 1] | [363, 58] | have ga : HolomorphicAt II I (uncurry g) (c, z) := by
have e : uncurry g = f ∘ fun p ↦ p.2 := rfl; rw [e]
apply HolomorphicAt.comp_of_eq fa holomorphicAt_snd; simp only | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, z)
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_ne_zero_eventually | [355, 1] | [363, 58] | have pc : Tendsto (fun z ↦ (c, z)) (𝓝 z) (𝓝 (c, z)) := continuousAt_const.prod continuousAt_id | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, z)
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, z)
pc : Tendsto (fun z => (c, z)) (𝓝 z) (𝓝 (c, z))
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, z)
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_ne_zero_eventually | [355, 1] | [363, 58] | exact pc.eventually (mfderiv_ne_zero_eventually' ga f0) | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, z)
pc : Tendsto (fun z => (c, z)) (𝓝 z) (𝓝 (c, z))
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, z)
pc : Tendsto (fun z => (c, z)) (𝓝 z) (𝓝 (c, z))
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_ne_zero_eventually | [355, 1] | [363, 58] | have e : uncurry g = f ∘ fun p ↦ p.2 := rfl | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, z) | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, z) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_ne_zero_eventually | [355, 1] | [363, 58] | rw [e] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, z) | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ HolomorphicAt (I.prod I) I (f ∘ fun p => p.2) (c, z) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_ne_zero_eventually | [355, 1] | [363, 58] | apply HolomorphicAt.comp_of_eq fa holomorphicAt_snd | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ HolomorphicAt (I.prod I) I (f ∘ fun p => p.2) (c, z) | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ (c, z).2 = z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ HolomorphicAt (I.prod I) I (f ∘ fun p => p.2) (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_ne_zero_eventually | [355, 1] | [363, 58] | simp only | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ (c, z).2 = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ (c, z).2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | isOpen_noncritical | [366, 1] | [368, 89] | rw [isOpen_iff_eventually] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ IsOpen {p | ¬Critical (f p.1) p.2} | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ ∀ x ∈ {p | ¬Critical (f p.1) p.2}, ∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | ¬Critical (f p.1) p.2} | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ IsOpen {p | ¬Critical (f p.1) p.2}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | isOpen_noncritical | [366, 1] | [368, 89] | intro ⟨c, z⟩ m | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ ∀ x ∈ {p | ¬Critical (f p.1) p.2}, ∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | ¬Critical (f p.1) p.2} | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : ℂ
z : S
m : (c, z) ∈ {p | ¬Critical (f p.1) p.2}
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | ¬Critical (f p.1) p.2} | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ ∀ x ∈ {p | ¬Critical (f p.1) p.2}, ∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | ¬Critical (f p.1) p.2}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | isOpen_noncritical | [366, 1] | [368, 89] | exact mfderiv_ne_zero_eventually' (fa _) m | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : ℂ
z : S
m : (c, z) ∈ {p | ¬Critical (f p.1) p.2}
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | ¬Critical (f p.1) p.2} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : ℂ
z : S
m : (c, z) ∈ {p | ¬Critical (f p.1) p.2}
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | ¬Critical (f p.1) p.2}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | isClosed_critical | [371, 1] | [374, 49] | have c := (isOpen_noncritical fa).isClosed_compl | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ IsClosed {p | Critical (f p.1) p.2} | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : IsClosed {p | ¬Critical (f p.1) p.2}ᶜ
⊢ IsClosed {p | Critical (f p.1) p.2} | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ IsClosed {p | Critical (f p.1) p.2}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | isClosed_critical | [371, 1] | [374, 49] | simp only [compl_setOf, not_not] at c | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : IsClosed {p | ¬Critical (f p.1) p.2}ᶜ
⊢ IsClosed {p | Critical (f p.1) p.2} | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : IsClosed {a | Critical (f a.1) a.2}
⊢ IsClosed {p | Critical (f p.1) p.2} | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : IsClosed {p | ¬Critical (f p.1) p.2}ᶜ
⊢ IsClosed {p | Critical (f p.1) p.2}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | isClosed_critical | [371, 1] | [374, 49] | exact c | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : IsClosed {a | Critical (f a.1) a.2}
⊢ IsClosed {p | Critical (f p.1) p.2} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : IsClosed {a | Critical (f a.1) a.2}
⊢ IsClosed {p | Critical (f p.1) p.2}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | rw [holomorphic_iff] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ Holomorphic (I.prod I) I f | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ Continuous f ∧
∀ (x : S × T),
AnalyticAt ℂ (↑(extChartAt I (f x)) ∘ f ∘ ↑(extChartAt (I.prod I) x).symm) (↑(extChartAt (I.prod I) x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ Holomorphic (I.prod I) I f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | use fc | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ Continuous f ∧
∀ (x : S × T),
AnalyticAt ℂ (↑(extChartAt I (f x)) ∘ f ∘ ↑(extChartAt (I.prod I) x).symm) (↑(extChartAt (I.prod I) x) x) | case right
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ ∀ (x : S × T),
AnalyticAt ℂ (↑(extChartAt I (f x)) ∘ f ∘ ↑(extChartAt (I.prod I) x).symm) (↑(extChartAt (I.prod I) x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ Continuous f ∧
∀ (x : S × T),
AnalyticAt ℂ (↑(extChartAt I (f x)) ∘ f ∘ ↑(extChartAt (I.prod I) x).symm) (↑(extChartAt (I.prod I) x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | intro p | case right
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ ∀ (x : S × T),
AnalyticAt ℂ (↑(extChartAt I (f x)) ∘ f ∘ ↑(extChartAt (I.prod I) x).symm) (↑(extChartAt (I.prod I) x) x) | case right
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ AnalyticAt ℂ (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (↑(extChartAt (I.prod I) p) p) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ ∀ (x : S × T),
AnalyticAt ℂ (↑(extChartAt I (f x)) ∘ f ∘ ↑(extChartAt (I.prod I) x).symm) (↑(extChartAt (I.prod I) x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | apply osgood_at' | case right
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ AnalyticAt ℂ (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (↑(extChartAt (I.prod I) p) p) | case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ AnalyticAt ℂ (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (↑(extChartAt (I.prod I) p) p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | have fm : ∀ᶠ q in 𝓝 (extChartAt II p p),
f ((extChartAt II p).symm q) ∈ (extChartAt I (f p)).source := by
refine (fc.continuousAt.comp (continuousAt_extChartAt_symm II p)).eventually_mem
((isOpen_extChartAt_source I (f p)).mem_nhds ?_)
simp only [Function.comp, (extChartAt II p).left_inv (mem_extChartAt_source _ _)]
apply mem_extChartAt_source | case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2 | case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | apply ((isOpen_extChartAt_target II p).eventually_mem (mem_extChartAt_target II p)).mp | case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2 | case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
x ∈ (extChartAt (I.prod I) p).target →
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | refine fm.mp (eventually_of_forall fun q fm m ↦ ⟨?_, ?_, ?_⟩) | case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
x ∈ (extChartAt (I.prod I) p).target →
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2 | case right.h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) q
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
x ∈ (extChartAt (I.prod I) p).target →
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | refine (fc.continuousAt.comp (continuousAt_extChartAt_symm II p)).eventually_mem
((isOpen_extChartAt_source I (f p)).mem_nhds ?_) | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ ∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ (f ∘ ↑(extChartAt (I.prod I) p).symm) (↑(extChartAt (I.prod I) p) p) ∈ (extChartAt I (f p)).source | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ ∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | simp only [Function.comp, (extChartAt II p).left_inv (mem_extChartAt_source _ _)] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ (f ∘ ↑(extChartAt (I.prod I) p).symm) (↑(extChartAt (I.prod I) p) p) ∈ (extChartAt I (f p)).source | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ f p ∈ (extChartAt I (f p)).source | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ (f ∘ ↑(extChartAt (I.prod I) p).symm) (↑(extChartAt (I.prod I) p) p) ∈ (extChartAt I (f p)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | apply mem_extChartAt_source | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ f p ∈ (extChartAt I (f p)).source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ f p ∈ (extChartAt I (f p)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | exact (continuousAt_extChartAt' I fm).comp_of_eq
(fc.continuousAt.comp (continuousAt_extChartAt_symm'' _ m)) rfl | case right.h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | apply HolomorphicAt.analyticAt I I | case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1 | case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | refine (HolomorphicAt.extChartAt fm).comp_of_eq ?_ rfl | case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1 | case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | rw [extChartAt_prod] at m | case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1 | case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ ((extChartAt I p.1).prod (extChartAt I p.2)).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | simp only [Function.comp, extChartAt_prod, PartialEquiv.prod_symm, PartialEquiv.prod_coe,
PartialEquiv.prod_target, mem_prod_eq] at m ⊢ | case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ ((extChartAt I p.1).prod (extChartAt I p.2)).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1 | case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q.1 ∈ (extChartAt I p.1).target ∧ q.2 ∈ (extChartAt I p.2).target
⊢ HolomorphicAt I I (fun z => f (↑(extChartAt I p.1).symm z, ↑(extChartAt I p.2).symm q.2)) q.1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ ((extChartAt I p.1).prod (extChartAt I p.2)).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | exact (f0 _ _).comp (HolomorphicAt.extChartAt_symm m.1) | case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q.1 ∈ (extChartAt I p.1).target ∧ q.2 ∈ (extChartAt I p.2).target
⊢ HolomorphicAt I I (fun z => f (↑(extChartAt I p.1).symm z, ↑(extChartAt I p.2).symm q.2)) q.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q.1 ∈ (extChartAt I p.1).target ∧ q.2 ∈ (extChartAt I p.2).target
⊢ HolomorphicAt I I (fun z => f (↑(extChartAt I p.1).symm z, ↑(extChartAt I p.2).symm q.2)) q.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | apply HolomorphicAt.analyticAt I I | case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2 | case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | refine (HolomorphicAt.extChartAt fm).comp_of_eq ?_ rfl | case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2 | case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | rw [extChartAt_prod] at m | case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2 | case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ ((extChartAt I p.1).prod (extChartAt I p.2)).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | simp only [Function.comp, extChartAt_prod, PartialEquiv.prod_symm, PartialEquiv.prod_coe,
PartialEquiv.prod_target, mem_prod_eq] at m ⊢ | case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ ((extChartAt I p.1).prod (extChartAt I p.2)).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2 | case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q.1 ∈ (extChartAt I p.1).target ∧ q.2 ∈ (extChartAt I p.2).target
⊢ HolomorphicAt I I (fun z => f (↑(extChartAt I p.1).symm q.1, ↑(extChartAt I p.2).symm z)) q.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ ((extChartAt I p.1).prod (extChartAt I p.2)).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | osgoodManifold | [379, 1] | [404, 60] | exact (f1 _ _).comp (HolomorphicAt.extChartAt_symm m.2) | case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q.1 ∈ (extChartAt I p.1).target ∧ q.2 ∈ (extChartAt I p.2).target
⊢ HolomorphicAt I I (fun z => f (↑(extChartAt I p.1).symm q.1, ↑(extChartAt I p.2).symm z)) q.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q.1 ∈ (extChartAt I p.1).target ∧ q.2 ∈ (extChartAt I p.2).target
⊢ HolomorphicAt I I (fun z => f (↑(extChartAt I p.1).symm q.1, ↑(extChartAt I p.2).symm z)) q.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_ofNat_lt | [14, 1] | [18, 84] | rw [←Real.exp_one_pow a] | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (2.7182818286 ^ a < b) _auto✝
⊢ (↑a).exp < b | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (2.7182818286 ^ a < b) _auto✝
⊢ exp 1 ^ a < b | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (2.7182818286 ^ a < b) _auto✝
⊢ (↑a).exp < b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_ofNat_lt | [14, 1] | [18, 84] | exact _root_.trans (pow_lt_pow_left Real.exp_one_lt_d9 (Real.exp_nonneg _) a0) h0 | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (2.7182818286 ^ a < b) _auto✝
⊢ exp 1 ^ a < b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (2.7182818286 ^ a < b) _auto✝
⊢ exp 1 ^ a < b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_neg_ofNat_lt | [20, 1] | [26, 13] | rw [Real.exp_neg, inv_lt, ←Real.exp_one_pow a] | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ (-↑a).exp < b | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ b⁻¹ < exp 1 ^ a
case ha
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ 0 < (↑a).exp
case hb
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ 0 < b | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ (-↑a).exp < b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_neg_ofNat_lt | [20, 1] | [26, 13] | exact _root_.trans h0 (pow_lt_pow_left Real.exp_one_gt_d9 (by norm_num) a0) | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ b⁻¹ < exp 1 ^ a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ b⁻¹ < exp 1 ^ a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_neg_ofNat_lt | [20, 1] | [26, 13] | norm_num | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ 0 ≤ 2.7182818283 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ 0 ≤ 2.7182818283
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_neg_ofNat_lt | [20, 1] | [26, 13] | exact Real.exp_pos _ | case ha
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ 0 < (↑a).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ 0 < (↑a).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_neg_ofNat_lt | [20, 1] | [26, 13] | exact b0 | case hb
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ 0 < b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ 0 < b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_ofNat | [28, 1] | [32, 78] | rw [←Real.exp_one_pow a] | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (b < 2.7182818283 ^ a) _auto✝
⊢ b < (↑a).exp | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (b < 2.7182818283 ^ a) _auto✝
⊢ b < exp 1 ^ a | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (b < 2.7182818283 ^ a) _auto✝
⊢ b < (↑a).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_ofNat | [28, 1] | [32, 78] | exact _root_.trans h0 (pow_lt_pow_left Real.exp_one_gt_d9 (by norm_num) a0) | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (b < 2.7182818283 ^ a) _auto✝
⊢ b < exp 1 ^ a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (b < 2.7182818283 ^ a) _auto✝
⊢ b < exp 1 ^ a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_ofNat | [28, 1] | [32, 78] | norm_num | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (b < 2.7182818283 ^ a) _auto✝
⊢ 0 ≤ 2.7182818283 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (b < 2.7182818283 ^ a) _auto✝
⊢ 0 ≤ 2.7182818283
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | have e : exp (a/b : ℝ) = exp 1 ^ (a / b : ℝ) := by rw [Real.exp_one_rpow] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
⊢ c < (↑a / ↑b).exp | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ c < (↑a / ↑b).exp | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
⊢ c < (↑a / ↑b).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | rw [e, div_eq_mul_inv, Real.rpow_mul (Real.exp_nonneg _), Real.lt_rpow_inv_iff_of_pos,
Real.rpow_natCast, Real.rpow_natCast] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ c < (↑a / ↑b).exp | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ c ^ b < exp 1 ^ a
case hx
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ c
case hy
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ exp 1 ^ ↑a
case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 < ↑b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ c < (↑a / ↑b).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | rw [Real.exp_one_rpow] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
⊢ (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
⊢ (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | exact _root_.trans h0 (pow_lt_pow_left Real.exp_one_gt_d9 (by norm_num) a0) | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ c ^ b < exp 1 ^ a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ c ^ b < exp 1 ^ a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | norm_num | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ 2.7182818283 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ 2.7182818283
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | exact c0.le | case hx
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | apply Real.rpow_nonneg (Real.exp_nonneg _) | case hy
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ exp 1 ^ ↑a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hy
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ exp 1 ^ ↑a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | simp only [Nat.cast_pos, Nat.pos_iff_ne_zero] | case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 < ↑b | case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ b ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 < ↑b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | exact b0 | case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ b ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ b ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | have e : exp (a/b : ℝ) = exp 1 ^ (a / b : ℝ) := by rw [Real.exp_one_rpow] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
⊢ (↑a / ↑b).exp < c | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ (↑a / ↑b).exp < c | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
⊢ (↑a / ↑b).exp < c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | rw [e, div_eq_mul_inv, Real.rpow_mul (Real.exp_nonneg _), Real.rpow_inv_lt_iff_of_pos,
Real.rpow_natCast, Real.rpow_natCast] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ (↑a / ↑b).exp < c | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ exp 1 ^ a < c ^ b
case hx
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ exp 1 ^ ↑a
case hy
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ c
case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 < ↑b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ (↑a / ↑b).exp < c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | rw [Real.exp_one_rpow] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
⊢ (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
⊢ (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | exact _root_.trans (pow_lt_pow_left Real.exp_one_lt_d9 (Real.exp_nonneg _) a0) h0 | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ exp 1 ^ a < c ^ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ exp 1 ^ a < c ^ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | apply Real.rpow_nonneg (Real.exp_nonneg _) | case hx
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ exp 1 ^ ↑a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ exp 1 ^ ↑a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | exact c0.le | case hy
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hy
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | simp only [Nat.cast_pos, Nat.pos_iff_ne_zero] | case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 < ↑b | case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ b ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 < ↑b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | exact b0 | case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ b ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ b ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_neg_div_lt | [58, 1] | [63, 45] | rw [Real.exp_neg, inv_lt (Real.exp_pos _) c0] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c⁻¹ ^ b < 2.7182818283 ^ a) _auto✝
⊢ (-(↑a / ↑b)).exp < c | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c⁻¹ ^ b < 2.7182818283 ^ a) _auto✝
⊢ c⁻¹ < (↑a / ↑b).exp | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c⁻¹ ^ b < 2.7182818283 ^ a) _auto✝
⊢ (-(↑a / ↑b)).exp < c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_neg_div_lt | [58, 1] | [63, 45] | exact lt_exp_div a0 b0 (inv_pos.mpr c0) h0 | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c⁻¹ ^ b < 2.7182818283 ^ a) _auto✝
⊢ c⁻¹ < (↑a / ↑b).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c⁻¹ ^ b < 2.7182818283 ^ a) _auto✝
⊢ c⁻¹ < (↑a / ↑b).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_neg_div | [65, 1] | [70, 45] | rw [Real.exp_neg, lt_inv c0 (Real.exp_pos _)] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c⁻¹ ^ b) _auto✝
⊢ c < (-(↑a / ↑b)).exp | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c⁻¹ ^ b) _auto✝
⊢ (↑a / ↑b).exp < c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c⁻¹ ^ b) _auto✝
⊢ c < (-(↑a / ↑b)).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_neg_div | [65, 1] | [70, 45] | exact exp_div_lt a0 b0 (inv_pos.mpr c0) h0 | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c⁻¹ ^ b) _auto✝
⊢ (↑a / ↑b).exp < c⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c⁻¹ ^ b) _auto✝
⊢ (↑a / ↑b).exp < c⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_2 | [77, 1] | [79, 29] | rw [Real.lt_log_iff_exp_lt (by norm_num)] | ⊢ 0.693 < log 2 | ⊢ exp 0.693 < 2 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0.693 < log 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_2 | [77, 1] | [79, 29] | norm_num | ⊢ exp 0.693 < 2 | ⊢ (693 / 1000).exp < 2 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ exp 0.693 < 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_2 | [77, 1] | [79, 29] | exact exp_div_lt | ⊢ (693 / 1000).exp < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ (693 / 1000).exp < 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_2 | [77, 1] | [79, 29] | norm_num | ⊢ 0 < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0 < 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_3 | [80, 1] | [82, 29] | rw [Real.lt_log_iff_exp_lt (by norm_num)] | ⊢ 1.098 < log 3 | ⊢ exp 1.098 < 3 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 1.098 < log 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_3 | [80, 1] | [82, 29] | norm_num | ⊢ exp 1.098 < 3 | ⊢ (549 / 500).exp < 3 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ exp 1.098 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_3 | [80, 1] | [82, 29] | exact exp_div_lt | ⊢ (549 / 500).exp < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ (549 / 500).exp < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_3 | [80, 1] | [82, 29] | norm_num | ⊢ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_3_lt | [83, 1] | [85, 29] | rw [Real.log_lt_iff_lt_exp (by norm_num)] | ⊢ log 3 < 1.099 | ⊢ 3 < exp 1.099 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ log 3 < 1.099
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_3_lt | [83, 1] | [85, 29] | norm_num | ⊢ 3 < exp 1.099 | ⊢ 3 < (1099 / 1000).exp | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 3 < exp 1.099
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_3_lt | [83, 1] | [85, 29] | exact lt_exp_div | ⊢ 3 < (1099 / 1000).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 3 < (1099 / 1000).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_3_lt | [83, 1] | [85, 29] | norm_num | ⊢ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_4 | [86, 1] | [88, 29] | rw [Real.lt_log_iff_exp_lt (by norm_num)] | ⊢ 1.386 < log 4 | ⊢ exp 1.386 < 4 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 1.386 < log 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_4 | [86, 1] | [88, 29] | norm_num | ⊢ exp 1.386 < 4 | ⊢ (693 / 500).exp < 4 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ exp 1.386 < 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_4 | [86, 1] | [88, 29] | exact exp_div_lt | ⊢ (693 / 500).exp < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ (693 / 500).exp < 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_4 | [86, 1] | [88, 29] | norm_num | ⊢ 0 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0 < 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_4_lt | [89, 1] | [91, 29] | rw [Real.log_lt_iff_lt_exp (by norm_num)] | ⊢ log 4 < 1.387 | ⊢ 4 < exp 1.387 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ log 4 < 1.387
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_4_lt | [89, 1] | [91, 29] | norm_num | ⊢ 4 < exp 1.387 | ⊢ 4 < (1387 / 1000).exp | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 4 < exp 1.387
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_4_lt | [89, 1] | [91, 29] | exact lt_exp_div | ⊢ 4 < (1387 / 1000).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 4 < (1387 / 1000).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_4_lt | [89, 1] | [91, 29] | norm_num | ⊢ 0 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0 < 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | rw [neg_lt, Real.lt_log_iff_exp_lt (lt_of_lt_of_le (by norm_num) le)] | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ -(1 - 1 / x).log < 0.41 | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-0.41).exp < 1 - 1 / x | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ -(1 - 1 / x).log < 0.41
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | refine lt_of_lt_of_le ?_ le | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-0.41).exp < 1 - 1 / x | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-0.41).exp < 2 / 3 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-0.41).exp < 1 - 1 / x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | norm_num | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-0.41).exp < 2 / 3 | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-(41 / 100)).exp < 2 / 3 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-0.41).exp < 2 / 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | exact exp_neg_div_lt | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-(41 / 100)).exp < 2 / 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-(41 / 100)).exp < 2 / 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | rw [le_tsub_iff_le_tsub (by norm_num), one_div, inv_le (by positivity)] | x : ℝ
x3 : 3 ≤ x
⊢ 2 / 3 ≤ 1 - 1 / x | x : ℝ
x3 : 3 ≤ x
⊢ (1 - 2 / 3)⁻¹ ≤ x
x : ℝ
x3 : 3 ≤ x
⊢ 0 < 1 - 2 / 3
x : ℝ
x3 : 3 ≤ x
⊢ 1 / x ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
⊢ 2 / 3 ≤ 1 - 1 / x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | norm_num | x : ℝ
x3 : 3 ≤ x
⊢ 2 / 3 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
⊢ 2 / 3 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | positivity | x : ℝ
x3 : 3 ≤ x
⊢ 0 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
⊢ 0 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | exact le_trans (by norm_num) x3 | x : ℝ
x3 : 3 ≤ x
⊢ (1 - 2 / 3)⁻¹ ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
⊢ (1 - 2 / 3)⁻¹ ≤ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | norm_num | x : ℝ
x3 : 3 ≤ x
⊢ (1 - 2 / 3)⁻¹ ≤ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
⊢ (1 - 2 / 3)⁻¹ ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | norm_num | x : ℝ
x3 : 3 ≤ x
⊢ 0 < 1 - 2 / 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
⊢ 0 < 1 - 2 / 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | rw [one_div, inv_le_one_iff] | x : ℝ
x3 : 3 ≤ x
⊢ 1 / x ≤ 1 | x : ℝ
x3 : 3 ≤ x
⊢ x ≤ 0 ∨ 1 ≤ x | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
⊢ 1 / x ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | right | x : ℝ
x3 : 3 ≤ x
⊢ x ≤ 0 ∨ 1 ≤ x | case h
x : ℝ
x3 : 3 ≤ x
⊢ 1 ≤ x | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
⊢ x ≤ 0 ∨ 1 ≤ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | exact le_trans (by norm_num) x3 | case h
x : ℝ
x3 : 3 ≤ x
⊢ 1 ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x : ℝ
x3 : 3 ≤ x
⊢ 1 ≤ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | norm_num | x : ℝ
x3 : 3 ≤ x
⊢ 1 ≤ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
⊢ 1 ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | norm_num | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ 0 < 2 / 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ 0 < 2 / 3
TACTIC:
|
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