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/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | use convex_univ | case hf
⊢ ConvexOn ℝ univ fun p => p.1 | case right
⊢ ∀ ⦃x : ℝ × ℝ⦄,
x ∈ univ → ∀ ⦃y : ℝ × ℝ⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y).1 ≤ a • x.1 + b • y.1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
⊢ ConvexOn ℝ univ fun p => p.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | intros | case right
⊢ ∀ ⦃x : ℝ × ℝ⦄,
x ∈ univ → ∀ ⦃y : ℝ × ℝ⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y).1 ≤ a • x.1 + b • y.1 | case right
x✝ : ℝ × ℝ
a✝⁵ : x✝ ∈ univ
y✝ : ℝ × ℝ
a✝⁴ : y✝ ∈ univ
a✝³ b✝ : ℝ
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ (a✝³ • x✝ + b✝ • y✝).1 ≤ a✝³ • x✝.1 + b✝ • y✝.1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
⊢ ∀ ⦃x : ℝ × ℝ⦄,
x ∈ univ → ∀ ⦃y : ℝ × ℝ⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y).1 ≤ a • x.1 + b • y.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | simp | case right
x✝ : ℝ × ℝ
a✝⁵ : x✝ ∈ univ
y✝ : ℝ × ℝ
a✝⁴ : y✝ ∈ univ
a✝³ b✝ : ℝ
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ (a✝³ • x✝ + b✝ • y✝).1 ≤ a✝³ • x✝.1 + b✝ • y✝.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
x✝ : ℝ × ℝ
a✝⁵ : x✝ ∈ univ
y✝ : ℝ × ℝ
a✝⁴ : y✝ ∈ univ
a✝³ b✝ : ℝ
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ (a✝³ • x✝ + b✝ • y✝).1 ≤ a✝³ • x✝.1 + b✝ • y✝.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | use convex_univ | case hg
⊢ ConvexOn ℝ univ fun p => p.2 | case right
⊢ ∀ ⦃x : ℝ × ℝ⦄,
x ∈ univ → ∀ ⦃y : ℝ × ℝ⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y).2 ≤ a • x.2 + b • y.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
⊢ ConvexOn ℝ univ fun p => p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | intros | case right
⊢ ∀ ⦃x : ℝ × ℝ⦄,
x ∈ univ → ∀ ⦃y : ℝ × ℝ⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y).2 ≤ a • x.2 + b • y.2 | case right
x✝ : ℝ × ℝ
a✝⁵ : x✝ ∈ univ
y✝ : ℝ × ℝ
a✝⁴ : y✝ ∈ univ
a✝³ b✝ : ℝ
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ (a✝³ • x✝ + b✝ • y✝).2 ≤ a✝³ • x✝.2 + b✝ • y✝.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
⊢ ∀ ⦃x : ℝ × ℝ⦄,
x ∈ univ → ∀ ⦃y : ℝ × ℝ⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y).2 ≤ a • x.2 + b • y.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | simp | case right
x✝ : ℝ × ℝ
a✝⁵ : x✝ ∈ univ
y✝ : ℝ × ℝ
a✝⁴ : y✝ ∈ univ
a✝³ b✝ : ℝ
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ (a✝³ • x✝ + b✝ • y✝).2 ≤ a✝³ • x✝.2 + b✝ • y✝.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
x✝ : ℝ × ℝ
a✝⁵ : x✝ ∈ univ
y✝ : ℝ × ℝ
a✝⁴ : y✝ ∈ univ
a✝³ b✝ : ℝ
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ (a✝³ • x✝ + b✝ • y✝).2 ≤ a✝³ • x✝.2 + b✝ • y✝.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | ContinuousOn.partialSups | [25, 1] | [28, 76] | induction' n with n h | A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s | case zero
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) 0) s
case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) (n + 1)) s | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | ContinuousOn.partialSups | [25, 1] | [28, 76] | simp [fc 0] | case zero
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) 0) s
case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) (n + 1)) s | case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) (n + 1)) s | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) 0) s
case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) (n + 1)) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | ContinuousOn.partialSups | [25, 1] | [28, 76] | simp | case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) (n + 1)) s | case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n ⊔ f (n + 1) x) s | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) (n + 1)) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | ContinuousOn.partialSups | [25, 1] | [28, 76] | exact ContinuousOn.max h (fc _) | case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n ⊔ f (n + 1) x) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n ⊔ f (n + 1) x) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | intro c0 c1 cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ swap '' s → AnalyticAt ℂ (fun z0 => (f ∘ swap) (z0, c1)) c0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z0 => (f ∘ swap) (z0, c1)) c0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ swap '' s → AnalyticAt ℂ (fun z0 => (f ∘ swap) (z0, c1)) c0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | simp only [Function.comp_apply, Prod.swap_prod_mk] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z0 => (f ∘ swap) (z0, c1)) c0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z0 => f (c1, z0)) c0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z0 => (f ∘ swap) (z0, c1)) c0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | rw [swap_mem] at cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z0 => f (c1, z0)) c0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c1, c0) ∈ s
⊢ AnalyticAt ℂ (fun z0 => f (c1, z0)) c0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z0 => f (c1, z0)) c0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | exact h.fa1 c1 c0 cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c1, c0) ∈ s
⊢ AnalyticAt ℂ (fun z0 => f (c1, z0)) c0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c1, c0) ∈ s
⊢ AnalyticAt ℂ (fun z0 => f (c1, z0)) c0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | intro c0 c1 cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ swap '' s → AnalyticAt ℂ (fun z1 => (f ∘ swap) (c0, z1)) c1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z1 => (f ∘ swap) (c0, z1)) c1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ swap '' s → AnalyticAt ℂ (fun z1 => (f ∘ swap) (c0, z1)) c1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | simp only [Function.comp_apply, Prod.swap_prod_mk] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z1 => (f ∘ swap) (c0, z1)) c1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z1 => f (z1, c0)) c1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z1 => (f ∘ swap) (c0, z1)) c1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | rw [swap_mem] at cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z1 => f (z1, c0)) c1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c1, c0) ∈ s
⊢ AnalyticAt ℂ (fun z1 => f (z1, c0)) c1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z1 => f (z1, c0)) c1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | exact h.fa0 c1 c0 cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c1, c0) ∈ s
⊢ AnalyticAt ℂ (fun z1 => f (z1, c0)) c1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c1, c0) ∈ s
⊢ AnalyticAt ℂ (fun z1 => f (z1, c0)) c1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on0 | [85, 1] | [90, 19] | intro z0 z0s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
⊢ AnalyticOn ℂ (fun z0 => f (z0, z1)) (closedBall c0 r) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
⊢ AnalyticOn ℂ (fun z0 => f (z0, z1)) (closedBall c0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on0 | [85, 1] | [90, 19] | apply h.fa0 z0 z1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ (z0, z1) ∈ closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on0 | [85, 1] | [90, 19] | rw [← closedBall_prod_same] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ (z0, z1) ∈ closedBall (c0, c1) r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ (z0, z1) ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ (z0, z1) ∈ closedBall (c0, c1) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on0 | [85, 1] | [90, 19] | simp only [Set.prod_mk_mem_set_prod_eq, Metric.mem_closedBall] at z0s ⊢ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ (z0, z1) ∈ closedBall c0 r ×ˢ closedBall c1 r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : dist z0 c0 ≤ r
⊢ dist z0 c0 ≤ r ∧ dist z1 c1 ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ (z0, z1) ∈ closedBall c0 r ×ˢ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on0 | [85, 1] | [90, 19] | exact ⟨z0s, z1r⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : dist z0 c0 ≤ r
⊢ dist z0 c0 ≤ r ∧ dist z1 c1 ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : dist z0 c0 ≤ r
⊢ dist z0 c0 ≤ r ∧ dist z1 c1 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on1 | [93, 1] | [98, 19] | intro z1 z1s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
⊢ AnalyticOn ℂ (fun z1 => f (z0, z1)) (closedBall c1 r) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
⊢ AnalyticOn ℂ (fun z1 => f (z0, z1)) (closedBall c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on1 | [93, 1] | [98, 19] | apply h.fa1 z0 z1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ (z0, z1) ∈ closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on1 | [93, 1] | [98, 19] | rw [← closedBall_prod_same] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ (z0, z1) ∈ closedBall (c0, c1) r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ (z0, z1) ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ (z0, z1) ∈ closedBall (c0, c1) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on1 | [93, 1] | [98, 19] | simp only [Set.prod_mk_mem_set_prod_eq, Metric.mem_closedBall] at z1s ⊢ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ (z0, z1) ∈ closedBall c0 r ×ˢ closedBall c1 r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : dist z1 c1 ≤ r
⊢ dist z0 c0 ≤ r ∧ dist z1 c1 ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ (z0, z1) ∈ closedBall c0 r ×ˢ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on1 | [93, 1] | [98, 19] | exact ⟨z0r, z1s⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : dist z1 c1 ≤ r
⊢ dist z0 c0 ≤ r ∧ dist z1 c1 ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : dist z1 c1 ≤ r
⊢ dist z0 c0 ≤ r ∧ dist z1 c1 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | generalize hu : min (r / 2) (e * r / b / 24) = u | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | rw [hu] at wz | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | have up : 0 < u := by
rw [← hu]; simp only [gt_iff_lt, lt_min_iff]
exact ⟨by bound, by bound⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | have ur : u ≤ r / 2 := by rw [← hu]; exact min_le_left _ _ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | have ue : 6 * b / r * u ≤ e / 4 := by
rw [← hu]
calc 6 * b / r * min (r / 2) (e * r / b / 24)
_ ≤ 6 * b / r * (e * r / b / 24) := by bound
_ = b / b * (r / r) * (e / 4) := by ring
_ = e / 4 := by field_simp [bp.ne', rp.ne'] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | rw [ball_prod_same'] at rs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | rw [← Metric.mem_ball, ball_prod_same', Set.mem_prod] at wz | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | have d : DifferentiableOn ℂ (fun t ↦ f (t, w.snd)) (ball z.fst (r / 2)) := by
intro t ts; refine (h.fa0 t w.snd ?_).differentiableWithinAt
exact rs (Set.mk_mem_prod ts (Metric.ball_subset_ball ur wz.right)) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | have wf : w.fst ∈ ball z.fst (r / 2) := Metric.ball_subset_ball ur wz.left | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | have m : Set.MapsTo (fun t ↦ f (t, w.snd)) (ball z.fst (r / 2))
(ball (f (z.fst, w.snd)) (3 * b)) := by
intro t ts; simp only [dist_eq_norm, Metric.mem_ball]; apply lt_of_le_of_lt (norm_sub_le _ _)
have f0 : ‖f (t, w.snd)‖ ≤ b := by
apply_rules [rs, Set.mk_mem_prod, Metric.ball_subset_ball ur, fb, wz.right]
have f1 : ‖f (z.fst, w.snd)‖ ≤ b := by
apply_rules [rs, Set.mk_mem_prod, Metric.ball_subset_ball ur, fb, Metric.mem_ball_self up,
wz.right]
calc ‖f (t, w.snd)‖ + ‖f (z.fst, w.snd)‖
_ ≤ b + b := by linarith
_ = 2 * b := by ring
_ < 3 * b := mul_lt_mul_of_pos_right (by norm_num) bp | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | have L := Complex.dist_le_div_mul_dist_of_mapsTo_ball d m wf | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f (w.1, w.2)) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | simp only [Prod.mk.eta] at L | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f (w.1, w.2)) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f (w.1, w.2)) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | refine _root_.trans L (_root_.trans ?_ ue) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
⊢ 3 * b / (r / 2) * dist w.1 z.1 ≤ 6 * b / r * u | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | simp only [Metric.mem_ball] at wz | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
⊢ 3 * b / (r / 2) * dist w.1 z.1 ≤ 6 * b / r * u | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
wz : dist w.1 z.1 < u ∧ dist w.2 z.2 < u
⊢ 3 * b / (r / 2) * dist w.1 z.1 ≤ 6 * b / r * u | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
⊢ 3 * b / (r / 2) * dist w.1 z.1 ≤ 6 * b / r * u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | rw [div_eq_mul_inv _ (2 : ℝ), div_mul_eq_div_div] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
wz : dist w.1 z.1 < u ∧ dist w.2 z.2 < u
⊢ 3 * b / (r / 2) * dist w.1 z.1 ≤ 6 * b / r * u | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
wz : dist w.1 z.1 < u ∧ dist w.2 z.2 < u
⊢ 3 * b / r / 2⁻¹ * dist w.1 z.1 ≤ 6 * b / r * u | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
wz : dist w.1 z.1 < u ∧ dist w.2 z.2 < u
⊢ 3 * b / (r / 2) * dist w.1 z.1 ≤ 6 * b / r * u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | ring_nf | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
wz : dist w.1 z.1 < u ∧ dist w.2 z.2 < u
⊢ 3 * b / r / 2⁻¹ * dist w.1 z.1 ≤ 6 * b / r * u | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
wz : dist w.1 z.1 < u ∧ dist w.2 z.2 < u
⊢ b * r⁻¹ * dist w.1 z.1 * 6 ≤ b * r⁻¹ * u * 6 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
wz : dist w.1 z.1 < u ∧ dist w.2 z.2 < u
⊢ 3 * b / r / 2⁻¹ * dist w.1 z.1 ≤ 6 * b / r * u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | bound | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
wz : dist w.1 z.1 < u ∧ dist w.2 z.2 < u
⊢ b * r⁻¹ * dist w.1 z.1 * 6 ≤ b * r⁻¹ * u * 6 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
m : Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
L : dist (f w) (f (z.1, w.2)) ≤ 3 * b / (r / 2) * dist w.1 z.1
wz : dist w.1 z.1 < u ∧ dist w.2 z.2 < u
⊢ b * r⁻¹ * dist w.1 z.1 * 6 ≤ b * r⁻¹ * u * 6
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | rw [← hu] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ 0 < u | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ 0 < min (r / 2) (e * r / b / 24) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ 0 < u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | simp only [gt_iff_lt, lt_min_iff] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ 0 < min (r / 2) (e * r / b / 24) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ 0 < r / 2 ∧ 0 < e * r / b / 24 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ 0 < min (r / 2) (e * r / b / 24)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | exact ⟨by bound, by bound⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ 0 < r / 2 ∧ 0 < e * r / b / 24 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ 0 < r / 2 ∧ 0 < e * r / b / 24
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | bound | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ 0 < r / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ 0 < r / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | bound | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ 0 < e * r / b / 24 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ 0 < e * r / b / 24
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | rw [← hu] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
⊢ u ≤ r / 2 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
⊢ min (r / 2) (e * r / b / 24) ≤ r / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
⊢ u ≤ r / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | exact min_le_left _ _ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
⊢ min (r / 2) (e * r / b / 24) ≤ r / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
⊢ min (r / 2) (e * r / b / 24) ≤ r / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | rw [← hu] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ 6 * b / r * u ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ 6 * b / r * min (r / 2) (e * r / b / 24) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ 6 * b / r * u ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | calc 6 * b / r * min (r / 2) (e * r / b / 24)
_ ≤ 6 * b / r * (e * r / b / 24) := by bound
_ = b / b * (r / r) * (e / 4) := by ring
_ = e / 4 := by field_simp [bp.ne', rp.ne'] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ 6 * b / r * min (r / 2) (e * r / b / 24) ≤ e / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ 6 * b / r * min (r / 2) (e * r / b / 24) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | bound | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ 6 * b / r * min (r / 2) (e * r / b / 24) ≤ 6 * b / r * (e * r / b / 24) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ 6 * b / r * min (r / 2) (e * r / b / 24) ≤ 6 * b / r * (e * r / b / 24)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | ring | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ 6 * b / r * (e * r / b / 24) = b / b * (r / r) * (e / 4) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ 6 * b / r * (e * r / b / 24) = b / b * (r / r) * (e / 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | field_simp [bp.ne', rp.ne'] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ b / b * (r / r) * (e / 4) = e / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
⊢ b / b * (r / r) * (e / 4) = e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | intro t ts | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
⊢ DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2)) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ DifferentiableWithinAt ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2)) t | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
⊢ DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | refine (h.fa0 t w.snd ?_).differentiableWithinAt | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ DifferentiableWithinAt ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2)) t | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ (t, w.2) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ DifferentiableWithinAt ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2)) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | exact rs (Set.mk_mem_prod ts (Metric.ball_subset_ball ur wz.right)) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ (t, w.2) ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ (t, w.2) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | intro t ts | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
⊢ Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b)) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ (fun t => f (t, w.2)) t ∈ ball (f (z.1, w.2)) (3 * b) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
⊢ Set.MapsTo (fun t => f (t, w.2)) (ball z.1 (r / 2)) (ball (f (z.1, w.2)) (3 * b))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | simp only [dist_eq_norm, Metric.mem_ball] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ (fun t => f (t, w.2)) t ∈ ball (f (z.1, w.2)) (3 * b) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ ‖f (t, w.2) - f (z.1, w.2)‖ < 3 * b | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ (fun t => f (t, w.2)) t ∈ ball (f (z.1, w.2)) (3 * b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | apply lt_of_le_of_lt (norm_sub_le _ _) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ ‖f (t, w.2) - f (z.1, w.2)‖ < 3 * b | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ ‖f (t, w.2)‖ + ‖f (z.1, w.2)‖ < 3 * b | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ ‖f (t, w.2) - f (z.1, w.2)‖ < 3 * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | have f0 : ‖f (t, w.snd)‖ ≤ b := by
apply_rules [rs, Set.mk_mem_prod, Metric.ball_subset_ball ur, fb, wz.right] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ ‖f (t, w.2)‖ + ‖f (z.1, w.2)‖ < 3 * b | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
⊢ ‖f (t, w.2)‖ + ‖f (z.1, w.2)‖ < 3 * b | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ ‖f (t, w.2)‖ + ‖f (z.1, w.2)‖ < 3 * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | have f1 : ‖f (z.fst, w.snd)‖ ≤ b := by
apply_rules [rs, Set.mk_mem_prod, Metric.ball_subset_ball ur, fb, Metric.mem_ball_self up,
wz.right] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
⊢ ‖f (t, w.2)‖ + ‖f (z.1, w.2)‖ < 3 * b | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
f1 : ‖f (z.1, w.2)‖ ≤ b
⊢ ‖f (t, w.2)‖ + ‖f (z.1, w.2)‖ < 3 * b | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
⊢ ‖f (t, w.2)‖ + ‖f (z.1, w.2)‖ < 3 * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | calc ‖f (t, w.snd)‖ + ‖f (z.fst, w.snd)‖
_ ≤ b + b := by linarith
_ = 2 * b := by ring
_ < 3 * b := mul_lt_mul_of_pos_right (by norm_num) bp | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
f1 : ‖f (z.1, w.2)‖ ≤ b
⊢ ‖f (t, w.2)‖ + ‖f (z.1, w.2)‖ < 3 * b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
f1 : ‖f (z.1, w.2)‖ ≤ b
⊢ ‖f (t, w.2)‖ + ‖f (z.1, w.2)‖ < 3 * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | apply_rules [rs, Set.mk_mem_prod, Metric.ball_subset_ball ur, fb, wz.right] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ ‖f (t, w.2)‖ ≤ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
⊢ ‖f (t, w.2)‖ ≤ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | apply_rules [rs, Set.mk_mem_prod, Metric.ball_subset_ball ur, fb, Metric.mem_ball_self up,
wz.right] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
⊢ ‖f (z.1, w.2)‖ ≤ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
⊢ ‖f (z.1, w.2)‖ ≤ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | linarith | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
f1 : ‖f (z.1, w.2)‖ ≤ b
⊢ ‖f (t, w.2)‖ + ‖f (z.1, w.2)‖ ≤ b + b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
f1 : ‖f (z.1, w.2)‖ ≤ b
⊢ ‖f (t, w.2)‖ + ‖f (z.1, w.2)‖ ≤ b + b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | ring | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
f1 : ‖f (z.1, w.2)‖ ≤ b
⊢ b + b = 2 * b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
f1 : ‖f (z.1, w.2)‖ ≤ b
⊢ b + b = 2 * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | norm_num | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
f1 : ‖f (z.1, w.2)‖ ≤ b
⊢ 2 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z.1 (r / 2) ×ˢ ball z.2 (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : w.1 ∈ ball z.1 u ∧ w.2 ∈ ball z.2 u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
ur : u ≤ r / 2
ue : 6 * b / r * u ≤ e / 4
d : DifferentiableOn ℂ (fun t => f (t, w.2)) (ball z.1 (r / 2))
wf : w.1 ∈ ball z.1 (r / 2)
t : ℂ
ts : t ∈ ball z.1 (r / 2)
f0 : ‖f (t, w.2)‖ ≤ b
f1 : ‖f (z.1, w.2)‖ ≤ b
⊢ 2 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | have wrs : ball w (r / 2) ⊆ s := by
refine _root_.trans ?_ rs; apply Metric.ball_subset_ball'
have rr := _root_.trans wz.le (min_le_left _ _)
trans r / 2 + r / 2; linarith; ring_nf; apply le_refl | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | have rs' : ball (swap w) (r / 2) ⊆ swap '' s := by rw [ball_swap]; exact Set.image_subset _ wrs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | have wz' : dist (swap z) (swap w) < min (r / 2) (e * r / b / 24) := by rwa [dist_swap, dist_comm] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
wz' : dist z.swap w.swap < min (r / 2) (e * r / b / 24)
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | have fb' : ∀ z, z ∈ swap '' s → ‖(f ∘ swap) z‖ ≤ b := fun z zs ↦ fb z.swap (swap_mem'.mp zs) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
wz' : dist z.swap w.swap < min (r / 2) (e * r / b / 24)
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
wz' : dist z.swap w.swap < min (r / 2) (e * r / b / 24)
fb' : ∀ z ∈ swap '' s, ‖(f ∘ swap) z‖ ≤ b
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
wz' : dist z.swap w.swap < min (r / 2) (e * r / b / 24)
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | have d' := Bounded.dist0 h.flip bp ep rp rs' wz' fb' | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
wz' : dist z.swap w.swap < min (r / 2) (e * r / b / 24)
fb' : ∀ z ∈ swap '' s, ‖(f ∘ swap) z‖ ≤ b
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
wz' : dist z.swap w.swap < min (r / 2) (e * r / b / 24)
fb' : ∀ z ∈ swap '' s, ‖(f ∘ swap) z‖ ≤ b
d' : dist ((f ∘ swap) z.swap) ((f ∘ swap) (w.swap.1, z.swap.2)) ≤ e / 4
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
wz' : dist z.swap w.swap < min (r / 2) (e * r / b / 24)
fb' : ∀ z ∈ swap '' s, ‖(f ∘ swap) z‖ ≤ b
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | simp only [Function.comp_apply, Prod.swap_swap, Prod.fst_swap, Prod.snd_swap,
Prod.swap_prod_mk] at d' | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
wz' : dist z.swap w.swap < min (r / 2) (e * r / b / 24)
fb' : ∀ z ∈ swap '' s, ‖(f ∘ swap) z‖ ≤ b
d' : dist ((f ∘ swap) z.swap) ((f ∘ swap) (w.swap.1, z.swap.2)) ≤ e / 4
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
wz' : dist z.swap w.swap < min (r / 2) (e * r / b / 24)
fb' : ∀ z ∈ swap '' s, ‖(f ∘ swap) z‖ ≤ b
d' : dist (f z) (f (z.1, w.2)) ≤ e / 4
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
wz' : dist z.swap w.swap < min (r / 2) (e * r / b / 24)
fb' : ∀ z ∈ swap '' s, ‖(f ∘ swap) z‖ ≤ b
d' : dist ((f ∘ swap) z.swap) ((f ∘ swap) (w.swap.1, z.swap.2)) ≤ e / 4
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | rwa [dist_comm] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
wz' : dist z.swap w.swap < min (r / 2) (e * r / b / 24)
fb' : ∀ z ∈ swap '' s, ‖(f ∘ swap) z‖ ≤ b
d' : dist (f z) (f (z.1, w.2)) ≤ e / 4
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
wz' : dist z.swap w.swap < min (r / 2) (e * r / b / 24)
fb' : ∀ z ∈ swap '' s, ‖(f ∘ swap) z‖ ≤ b
d' : dist (f z) (f (z.1, w.2)) ≤ e / 4
⊢ dist (f (z.1, w.2)) (f z) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | refine _root_.trans ?_ rs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ ball w (r / 2) ⊆ s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ ball w (r / 2) ⊆ ball z r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ ball w (r / 2) ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | apply Metric.ball_subset_ball' | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ ball w (r / 2) ⊆ ball z r | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ r / 2 + dist w z ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ ball w (r / 2) ⊆ ball z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | have rr := _root_.trans wz.le (min_le_left _ _) | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ r / 2 + dist w z ≤ r | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r / 2 + dist w z ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ r / 2 + dist w z ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | trans r / 2 + r / 2 | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r / 2 + dist w z ≤ r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r / 2 + dist w z ≤ r / 2 + r / 2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r / 2 + r / 2 ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r / 2 + dist w z ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | linarith | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r / 2 + dist w z ≤ r / 2 + r / 2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r / 2 + r / 2 ≤ r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r / 2 + r / 2 ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r / 2 + dist w z ≤ r / 2 + r / 2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r / 2 + r / 2 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | ring_nf | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r / 2 + r / 2 ≤ r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r / 2 + r / 2 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | apply le_refl | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
rr : dist w z ≤ r / 2
⊢ r ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | rw [ball_swap] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
⊢ ball w.swap (r / 2) ⊆ swap '' s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
⊢ swap '' ball w (r / 2) ⊆ swap '' s | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
⊢ ball w.swap (r / 2) ⊆ swap '' s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | exact Set.image_subset _ wrs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
⊢ swap '' ball w (r / 2) ⊆ swap '' s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
⊢ swap '' ball w (r / 2) ⊆ swap '' s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist1 | [139, 1] | [152, 18] | rwa [dist_swap, dist_comm] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
⊢ dist z.swap w.swap < min (r / 2) (e * r / b / 24) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : e > 0
rp : r > 0
rs : ball z r ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
wrs : ball w (r / 2) ⊆ s
rs' : ball w.swap (r / 2) ⊆ swap '' s
⊢ dist z.swap w.swap < min (r / 2) (e * r / b / 24)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | suffices c : ContinuousOn f s by exact osgood o c h.fa0 h.fa1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ AnalyticOn ℂ f s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ ContinuousOn f s | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ AnalyticOn ℂ f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | by_cases bp : b ≤ 0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ ContinuousOn f s | case pos
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : b ≤ 0
⊢ ContinuousOn f s
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : ¬b ≤ 0
⊢ ContinuousOn f s | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ ContinuousOn f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | simp only [not_le] at bp | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : ¬b ≤ 0
⊢ ContinuousOn f s | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
⊢ ContinuousOn f s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : ¬b ≤ 0
⊢ ContinuousOn f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | intro z zs | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
⊢ ContinuousOn f s | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
⊢ ContinuousWithinAt f s z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
⊢ ContinuousOn f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | rcases Metric.isOpen_iff.mp o z zs with ⟨r, rp, rs⟩ | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
⊢ ContinuousWithinAt f s z | case neg.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
⊢ ContinuousWithinAt f s z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
⊢ ContinuousWithinAt f s z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | rw [Metric.continuousWithinAt_iff] | case neg.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
⊢ ContinuousWithinAt f s z | case neg.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
⊢ ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ × ℂ}, x ∈ s → dist x z < δ → dist (f x) (f z) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
⊢ ContinuousWithinAt f s z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | intro e ep | case neg.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
⊢ ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ × ℂ}, x ∈ s → dist x z < δ → dist (f x) (f z) < ε | case neg.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
⊢ ∃ δ > 0, ∀ {x : ℂ × ℂ}, x ∈ s → dist x z < δ → dist (f x) (f z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
⊢ ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ × ℂ}, x ∈ s → dist x z < δ → dist (f x) (f z) < ε
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | have up : min (r / 2) (e * r / b / 24) > 0 := by bound | case neg.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
⊢ ∃ δ > 0, ∀ {x : ℂ × ℂ}, x ∈ s → dist x z < δ → dist (f x) (f z) < e | case neg.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
⊢ ∃ δ > 0, ∀ {x : ℂ × ℂ}, x ∈ s → dist x z < δ → dist (f x) (f z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
⊢ ∃ δ > 0, ∀ {x : ℂ × ℂ}, x ∈ s → dist x z < δ → dist (f x) (f z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | use min (r / 2) (e * r / b / 24), up | case neg.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
⊢ ∃ δ > 0, ∀ {x : ℂ × ℂ}, x ∈ s → dist x z < δ → dist (f x) (f z) < e | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
⊢ ∀ {x : ℂ × ℂ}, x ∈ s → dist x z < min (r / 2) (e * r / b / 24) → dist (f x) (f z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
⊢ ∃ δ > 0, ∀ {x : ℂ × ℂ}, x ∈ s → dist x z < δ → dist (f x) (f z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | intro w _ wz | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
⊢ ∀ {x : ℂ × ℂ}, x ∈ s → dist x z < min (r / 2) (e * r / b / 24) → dist (f x) (f z) < e | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : ℂ × ℂ
a✝ : w ∈ s
wz : dist w z < min (r / 2) (e * r / b / 24)
⊢ dist (f w) (f z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
⊢ ∀ {x : ℂ × ℂ}, x ∈ s → dist x z < min (r / 2) (e * r / b / 24) → dist (f x) (f z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | have s0 : dist (f w) (f (z.fst, w.snd)) ≤ e / 4 :=
Bounded.dist0 h bp ep rp (_root_.trans (Metric.ball_subset_ball (by linarith)) rs) wz fb | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : ℂ × ℂ
a✝ : w ∈ s
wz : dist w z < min (r / 2) (e * r / b / 24)
⊢ dist (f w) (f z) < e | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : ℂ × ℂ
a✝ : w ∈ s
wz : dist w z < min (r / 2) (e * r / b / 24)
s0 : dist (f w) (f (z.1, w.2)) ≤ e / 4
⊢ dist (f w) (f z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : ℂ × ℂ
a✝ : w ∈ s
wz : dist w z < min (r / 2) (e * r / b / 24)
⊢ dist (f w) (f z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | have s1 : dist (f (z.fst, w.snd)) (f z) ≤ e / 4 := Bounded.dist1 h bp ep rp rs wz fb | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : ℂ × ℂ
a✝ : w ∈ s
wz : dist w z < min (r / 2) (e * r / b / 24)
s0 : dist (f w) (f (z.1, w.2)) ≤ e / 4
⊢ dist (f w) (f z) < e | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : ℂ × ℂ
a✝ : w ∈ s
wz : dist w z < min (r / 2) (e * r / b / 24)
s0 : dist (f w) (f (z.1, w.2)) ≤ e / 4
s1 : dist (f (z.1, w.2)) (f z) ≤ e / 4
⊢ dist (f w) (f z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
o : IsOpen s
b : ℝ
fb : ∀ z ∈ s, ‖f z‖ ≤ b
bp : 0 < b
z : ℂ × ℂ
zs : z ∈ s
r : ℝ
rp : r > 0
rs : ball z r ⊆ s
e : ℝ
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : ℂ × ℂ
a✝ : w ∈ s
wz : dist w z < min (r / 2) (e * r / b / 24)
s0 : dist (f w) (f (z.1, w.2)) ≤ e / 4
⊢ dist (f w) (f z) < e
TACTIC:
|
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