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/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_four_le_same | [36, 1] | [42, 53] | calc volume.real (ball x (4 * r))
= volume.real (ball x (2 * (2 * r))) := by ring_nf
_ ≤ A * volume.real (ball x (2 * r)) := by apply volume_ball_two_le_same
_ ≤ A * (A * volume.real (ball x r)) := by gcongr; apply volume_ball_two_le_same
_ = A ^ 2 * volume.real (ball x r) := by ring_nf | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ volume.real (ball x (4 * r)) ≤ A ^ 2 * volume.real (ball x r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ volume.real (ball x (4 * r)) ≤ A ^ 2 * volume.real (ball x r)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_four_le_same | [36, 1] | [42, 53] | ring_nf | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ volume.real (ball x (4 * r)) = volume.real (ball x (2 * (2 * r))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ volume.real (ball x (4 * r)) = volume.real (ball x (2 * (2 * r)))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_four_le_same | [36, 1] | [42, 53] | apply volume_ball_two_le_same | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ volume.real (ball x (2 * (2 * r))) ≤ A * volume.real (ball x (2 * r)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ volume.real (ball x (2 * (2 * r))) ≤ A * volume.real (ball x (2 * r))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_four_le_same | [36, 1] | [42, 53] | gcongr | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ A * volume.real (ball x (2 * r)) ≤ A * (A * volume.real (ball x r)) | case h
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ volume.real (ball x (2 * r)) ≤ A * volume.real (ball x r) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ A * volume.real (ball x (2 * r)) ≤ A * (A * volume.real (ball x r))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_four_le_same | [36, 1] | [42, 53] | apply volume_ball_two_le_same | case h
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ volume.real (ball x (2 * r)) ≤ A * volume.real (ball x r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ volume.real (ball x (2 * r)) ≤ A * volume.real (ball x r)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_four_le_same | [36, 1] | [42, 53] | ring_nf | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ A * (A * volume.real (ball x r)) = A ^ 2 * volume.real (ball x r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r : ℝ
⊢ A * (A * volume.real (ball x r)) = A ^ 2 * volume.real (ball x r)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | have hn (n : ℕ) : volume.real (ball x (2^n * r)) ≤ A^n * volume.real (ball x r) := by
induction n
case zero =>
simp
case succ m hm =>
calc volume.real (ball x (2 ^ (Nat.succ m) * r))
= volume.real (ball x (2 ^ (m+1) * r)) := by rfl
_ = volume.real (ball x ((2 ^ m*2^1) * r)) := by norm_cast
... | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
⊢ volume.real (ball x r') ≤ As A s * volume.real (ball x r) | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
⊢ volume.real (ball x r') ≤ As A s * volume.real (ball x r) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
⊢ volume.real (ball x r') ≤ As A s * volume.real (ball x r)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | have haux : s * r ≤ 2 ^ ⌈Real.log s / Real.log 2⌉₊ * r := by
sorry | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
⊢ volume.real (ball x r') ≤ As A s * volume.real (ball x r) | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ * r
⊢ volume.real (ball x r') ≤ As A s * volume.real (ba... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
⊢ volume.real (ball x r') ≤ As A s *... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | have h1 : ball x r' ⊆ ball x (2 ^ ⌈Real.log s / Real.log 2⌉₊ * r) := by
calc ball x r' ⊆ ball x (s * r) := by apply ball_subset_ball hs
_ ⊆ ball x (2 ^ ⌈Real.log s / Real.log 2⌉₊ * r) := by apply ball_subset_ball haux | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ * r
⊢ volume.real (ball x r') ≤ As A s * volume.real (ba... | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ * r
h1 : ball x r' ⊆ ball x (2 ^ ⌈s.log / Real.log 2⌉₊ *... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | calc volume.real (ball x r')
≤ volume.real (ball x (2 ^ ⌈Real.log s / Real.log 2⌉₊ * r)) := by gcongr; finiteness
_ ≤ A^(⌈Real.log s / Real.log 2⌉₊) * volume.real (ball x r) := by apply hn | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ * r
h1 : ball x r' ⊆ ball x (2 ^ ⌈s.log / Real.log 2⌉₊ *... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | induction n | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
n : ℕ
⊢ volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r) | case zero
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
⊢ volume.real (ball x (2 ^ 0 * r)) ≤ A ^ 0 * volume.real (ball x r)
case succ
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousTyp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
n : ℕ
⊢ volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | case zero =>
simp | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
⊢ volume.real (ball x (2 ^ 0 * r)) ≤ A ^ 0 * volume.real (ball x r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
⊢ volume.real (ball x (2 ^ 0 * r)) ≤ A ^ 0 * volume.real (ball x r)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | case succ m hm =>
calc volume.real (ball x (2 ^ (Nat.succ m) * r))
= volume.real (ball x (2 ^ (m+1) * r)) := by rfl
_ = volume.real (ball x ((2 ^ m*2^1) * r)) := by norm_cast
_ = volume.real (ball x (2 * 2 ^ m * r)) := by ring_nf
_ ≤ A * volume.real (ball x (2 ^ m * r)) := by rw[mul_assoc]; norm_cas... | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 ^ (m + 1) * r)) ≤ A ^ (m + 1) * volume.real (ball x r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 ^ (m + 1) * r)) ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | simp | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
⊢ volume.real (ball x (2 ^ 0 * r)) ≤ A ^ 0 * volume.real (ball x r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
⊢ volume.real (ball x (2 ^ 0 * r)) ≤ A ^ 0 * volume.real (ball x r)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | calc volume.real (ball x (2 ^ (Nat.succ m) * r))
= volume.real (ball x (2 ^ (m+1) * r)) := by rfl
_ = volume.real (ball x ((2 ^ m*2^1) * r)) := by norm_cast
_ = volume.real (ball x (2 * 2 ^ m * r)) := by ring_nf
_ ≤ A * volume.real (ball x (2 ^ m * r)) := by rw[mul_assoc]; norm_cast; apply volume_ball_two_le_... | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 ^ (m + 1) * r)) ≤ A ^ (m + 1) * volume.real (ball x r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 ^ (m + 1) * r)) ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | rfl | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 ^ m.succ * r)) = volume.real (ball x (2 ^ (m + 1) * r)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 ^ m.succ * r)) =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | norm_cast | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 ^ (m + 1) * r)) = volume.real (ball x (2 ^ m * 2 ^ 1 * r)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 ^ (m + 1) * r)) ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | ring_nf | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 ^ m * 2 ^ 1 * r)) = volume.real (ball x (2 * 2 ^ m * r)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 ^ m * 2 ^ 1 * r)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | rw[mul_assoc] | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 * 2 ^ m * r)) ≤ A * volume.real (ball x (2 ^ m * r)) | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 * (2 ^ m * r))) ≤ A * volume.real (ball x (2 ^ m * r)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 * 2 ^ m * r)) ≤ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | norm_cast | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 * (2 ^ m * r))) ≤ A * volume.real (ball x (2 ^ m * r)) | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 * (↑(2 ^ m) * r))) ≤ A * volume.real (ball x (↑(2 ^ m) * r)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 * (2 ^ m * r))) ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | apply volume_ball_two_le_same | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 * (↑(2 ^ m) * r))) ≤ A * volume.real (ball x (↑(2 ^ m) * r)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ volume.real (ball x (2 * (↑(2 ^ m) * r)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | gcongr | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ A * volume.real (ball x (2 ^ m * r)) ≤ A * (A ^ m * volume.real (ball x r)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ A * volume.real (ball x (2 ^ m * r)) ≤ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | rw[<- mul_assoc, pow_succ'] | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ A * (A ^ m * volume.real (ball x r)) = A ^ m.succ * volume.real (ball x r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
m : ℕ
hm : volume.real (ball x (2 ^ m * r)) ≤ A ^ m * volume.real (ball x r)
⊢ A * (A ^ m * volume.real (ball x r)) = ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | sorry | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
⊢ s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ * r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
⊢ s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | calc ball x r' ⊆ ball x (s * r) := by apply ball_subset_ball hs
_ ⊆ ball x (2 ^ ⌈Real.log s / Real.log 2⌉₊ * r) := by apply ball_subset_ball haux | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ * r
⊢ ball x r' ⊆ ball x (2 ^ ⌈s.log / Real.log 2⌉₊ * r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | apply ball_subset_ball hs | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ * r
⊢ ball x r' ⊆ ball x (s * r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | apply ball_subset_ball haux | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ * r
⊢ ball x (s * r) ⊆ ball x (2 ^ ⌈s.log / Real.log 2⌉₊... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | gcongr | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ * r
h1 : ball x r' ⊆ ball x (2 ^ ⌈s.log / Real.log 2⌉₊ *... | case h₂
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ * r
h1 : ball x r' ⊆ ball x (2 ^ ⌈s.log / Real.l... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | finiteness | case h₂
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ * r
h1 : ball x r' ⊆ ball x (2 ^ ⌈s.log / Real.l... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_same | [53, 1] | [79, 83] | apply hn | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log 2⌉₊ * r
h1 : ball x r' ⊆ ball x (2 ^ ⌈s.log / Real.log 2⌉₊ *... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r s r' : ℝ
hsp : s > 0
hs : r' ≤ s * r
hn : ∀ (n : ℕ), volume.real (ball x (2 ^ n * r)) ≤ A ^ n * volume.real (ball x r)
haux : s * r ≤ 2 ^ ⌈s.log / Real.log... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | ball_subset_ball_of_le | [86, 1] | [93, 27] | intro y h | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
⊢ ball x' r' ⊆ ball x r | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ y ∈ ball x r | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
⊢ ball x' r' ⊆ ball x r
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | ball_subset_ball_of_le | [86, 1] | [93, 27] | have h1 : dist x y < r := by
calc dist x y ≤ dist x x' + dist x' y := by apply dist_triangle
_ < dist x x' + r' := by gcongr; apply mem_ball'.mp h
_ ≤ r := by apply hr | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ y ∈ ball x r | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
h1 : dist x y < r
⊢ y ∈ ball x r | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ y ∈ ball x r
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | ball_subset_ball_of_le | [86, 1] | [93, 27] | exact mem_ball'.mpr h1 | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
h1 : dist x y < r
⊢ y ∈ ball x r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
h1 : dist x y < r
⊢ y ∈ ball x r
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | ball_subset_ball_of_le | [86, 1] | [93, 27] | calc dist x y ≤ dist x x' + dist x' y := by apply dist_triangle
_ < dist x x' + r' := by gcongr; apply mem_ball'.mp h
_ ≤ r := by apply hr | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ dist x y < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ dist x y < r
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | ball_subset_ball_of_le | [86, 1] | [93, 27] | apply dist_triangle | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ dist x y ≤ dist x x' + dist x' y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ dist x y ≤ dist x x' + dist x' y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | ball_subset_ball_of_le | [86, 1] | [93, 27] | gcongr | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ dist x x' + dist x' y < dist x x' + r' | case bc
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ dist x' y < r' | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ dist x x' + dist x' y < dist x x' + r'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | ball_subset_ball_of_le | [86, 1] | [93, 27] | apply mem_ball'.mp h | case bc
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ dist x' y < r' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bc
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ dist x' y < r'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | ball_subset_ball_of_le | [86, 1] | [93, 27] | apply hr | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ dist x x' + r' ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' : ℝ
hr : dist x x' + r' ≤ r
y : X
h : y ∈ ball x' r'
⊢ dist x x' + r' ≤ r
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_of_dist_le | [96, 1] | [98, 77] | sorry | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' s d : ℝ
hs : r' ≤ s * r
hd : dist x x' ≤ d * r
⊢ volume.real (ball x' r') ≤ Ad A s d * volume.real (ball x r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x x' : X
r r' s d : ℝ
hs : r' ≤ s * r
hd : dist x x' ≤ d * r
⊢ volume.real (ball x' r') ≤ Ad A s d * volume.real (ball x r)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | volume_ball_le_of_subset | [102, 1] | [104, 80] | sorry | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x' x : X
r r' s : ℝ
hs : r' ≤ s * r
hr : ball x' r ⊆ ball x r'
⊢ volume.real (ball x (2 * r)) ≤ Ai A s * volume.real (ball x' r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x' x : X
r r' s : ℝ
hs : r' ≤ s * r
hr : ball x' r ⊆ ball x r'
⊢ volume.real (ball x (2 * r)) ≤ Ai A s * volume.real (ball x' r)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | card_le_of_le_dist | [116, 1] | [117, 99] | sorry | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r r' s : ℝ
P : Set X
hs : r' ≤ s * r
hP : P ⊆ ball x r'
h2P : ∀ (x y : X), x ∈ P → y ∈ P → x ≠ y → r ≤ dist x y
⊢ P.Finite ∧ Nat.card ↑P ≤ Np A s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
x : X
r r' s : ℝ
P : Set X
hs : r' ≤ s * r
hP : P ⊆ ball x r'
h2P : ∀ (x y : X), x ∈ P → y ∈ P → x ≠ y → r ≤ dist x y
⊢ P.Finite ∧ Nat.card ↑P ≤ Np A s
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/HomogeneousType.lean | ballsCoverBalls | [122, 1] | [123, 8] | sorry | X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
r r' s : ℝ
hs : r' ≤ s * r
⊢ BallsCoverBalls X r' r (Np A s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
hA : 1 ≤ A
inst✝¹ : PseudoMetricSpace X
inst✝ : IsSpaceOfHomogeneousType X A
r r' s : ℝ
hs : r' ≤ s * r
⊢ BallsCoverBalls X r' r (Np A s)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Proposition1.lean | prop2_1 | [34, 1] | [47, 75] | sorry | X : Type u_1
A : ℝ
inst✝⁶ : MetricSpace X
inst✝⁵ : IsSpaceOfHomogeneousType X A
inst✝⁴ : Inhabited X
τ q q' D κ C₀ C : ℝ
Θ : Set C(X, ℂ)
inst✝³ : IsCompatible Θ
inst✝² : IsCancellative τ Θ
inst✝¹ : TileStructure Θ D κ C₀
F G : Set X
σ σ' : X → ℤ
Q' : X → C(X, ℂ)
K : X → X → ℂ
inst✝ : IsCZKernel τ K
ψ : ℝ → ℝ
hA : 1 < A... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
inst✝⁶ : MetricSpace X
inst✝⁵ : IsSpaceOfHomogeneousType X A
inst✝⁴ : Inhabited X
τ q q' D κ C₀ C : ℝ
Θ : Set C(X, ℂ)
inst✝³ : IsCompatible Θ
inst✝² : IsCancellative τ Θ
inst✝¹ : TileStructure Θ D κ C₀
F G : Set X
σ σ' : X → ℤ
Q' : X → C(X,... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | rw [Metric.uniformContinuous_iff] | α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε | α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ↔
∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | constructor | α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ↔
∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε | case mp
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) →
∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
case mpr
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : Pseud... | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ↔
∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | . intro h ε εpos
obtain ⟨δ', δ'pos, hδ'⟩ := h ε εpos
use min δ' (b / 2)
constructor
. exact (lt_min δ'pos (by linarith)).gt
constructor
. apply min_lt_of_right_lt
linarith
. intro x y hxy
exact hδ' (lt_of_lt_of_le hxy (min_le_left δ' (b / 2))) | case mp
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) →
∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
case mpr
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : Pseud... | case mpr
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε) →
∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) →
∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
c... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | . intro h ε εpos
obtain ⟨δ, δpos, _, hδ⟩ := h ε εpos
use δ | case mpr
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε) →
∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε) →
∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
T... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | intro h ε εpos | case mp
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) →
∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε | case mp
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
⊢ ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) →
∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
TA... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | obtain ⟨δ', δ'pos, hδ'⟩ := h ε εpos | case mp
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
⊢ ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε | case mp.intro.intro
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dis... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
⊢ ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | use min δ' (b / 2) | case mp.intro.intro
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dis... | case h
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ min δ' (b / 2) > 0 ∧ min δ' (b / 2) < b ∧ ∀ {x... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | constructor | case h
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ min δ' (b / 2) > 0 ∧ min δ' (b / 2) < b ∧ ∀ {x... | case h.left
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ min δ' (b / 2) > 0
case h.right
α β : Ty... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | . exact (lt_min δ'pos (by linarith)).gt | case h.left
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ min δ' (b / 2) > 0
case h.right
α β : Ty... | case h.right
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ min δ' (b / 2) < b ∧ ∀ {x y : α}, dist x... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dis... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | constructor | case h.right
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ min δ' (b / 2) < b ∧ ∀ {x y : α}, dist x... | case h.right.left
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ min δ' (b / 2) < b
case h.right.ri... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → di... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | . apply min_lt_of_right_lt
linarith | case h.right.left
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ min δ' (b / 2) < b
case h.right.ri... | case h.right.right
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ ∀ {x y : α}, dist x y < min δ' (b ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ'... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | . intro x y hxy
exact hδ' (lt_of_lt_of_le hxy (min_le_left δ' (b / 2))) | case h.right.right
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ ∀ {x y : α}, dist x y < min δ' (b ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | exact (lt_min δ'pos (by linarith)).gt | case h.left
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ min δ' (b / 2) > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dis... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | linarith | α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ 0 < b / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | apply min_lt_of_right_lt | case h.right.left
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ min δ' (b / 2) < b | case h.right.left.h
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ b / 2 < b | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ'... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | linarith | case h.right.left.h
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ b / 2 < b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left.h
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | intro x y hxy | case h.right.right
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
⊢ ∀ {x y : α}, dist x y < min δ' (b ... | case h.right.right
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
x y : α
hxy : dist x y < min δ' (b /... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | exact hδ' (lt_of_lt_of_le hxy (min_le_left δ' (b / 2))) | case h.right.right
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ' → dist (f a) (f b) < ε
x y : α
hxy : dist x y < min δ' (b /... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
ε : ℝ
εpos : ε > 0
δ' : ℝ
δ'pos : δ' > 0
hδ' : ∀ {a b : α}, dist a b < δ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | intro h ε εpos | case mpr
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε) →
∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε | case mpr
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
ε : ℝ
εpos : ε > 0
⊢ ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
⊢ (∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε) →
∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε
T... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | obtain ⟨δ, δpos, _, hδ⟩ := h ε εpos | case mpr
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
ε : ℝ
εpos : ε > 0
⊢ ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε | case mpr.intro.intro.intro
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
left✝ : δ < b
hδ : ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
⊢ ∃ δ > 0, ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
ε : ℝ
εpos : ε > 0
⊢ ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | uniformContinuous_iff_bounded | [23, 1] | [39, 10] | use δ | case mpr.intro.intro.intro
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
left✝ : δ < b
hδ : ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
⊢ ∃ δ > 0, ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro
α β : Type
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
b : ℝ
bpos : b > 0
h : ∀ ε > 0, ∃ δ > 0, δ < b ∧ ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
left✝ : δ < b
hδ :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | obtain ⟨δ, δpos, hδ⟩ := (Metric.uniformContinuous_iff.mp unicontf) ε εpos | f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε | case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | let φ : ContDiffBump (0 : ℝ) := ⟨δ/2, δ, by linarith, by linarith⟩ | case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε | case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | let f_0 := MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLinearMap.lsmul ℝ ℝ) MeasureTheory.volume | case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε | case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (Continuous... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ ∀... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | use f_0 | case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (Continuous... | case h
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLinearMap.... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureThe... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | constructor | case h
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLinearMap.... | case h.left
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLinea... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convol... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | .
apply HasCompactSupport.contDiff_convolution_left
. exact ContDiffBump.hasCompactSupport_normed φ
. exact ContDiffBump.contDiff_normed φ
. refine Continuous.locallyIntegrable ?h.left.hg.hf
exact unicontf.continuous | case h.left
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLinea... | case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLine... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.c... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | . intro x
rw [← Complex.dist_eq, dist_comm]
apply ContDiffBump.dist_normed_convolution_le
. exact unicontf.continuous.aestronglyMeasurable
. intro y hy
simp only [Metric.mem_ball] at hy
exact (hδ hy).le | case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLine... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | linarith | f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ 0 < δ / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ 0 < δ / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | linarith | f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ δ / 2 < δ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ δ / 2 < δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | apply HasCompactSupport.contDiff_convolution_left | case h.left
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLinea... | case h.left.hcf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousL... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.c... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | . exact ContDiffBump.hasCompactSupport_normed φ | case h.left.hcf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousL... | case h.left.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hcf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheo... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | . exact ContDiffBump.contDiff_normed φ | case h.left.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLi... | case h.left.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLi... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheor... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | . refine Continuous.locallyIntegrable ?h.left.hg.hf
exact unicontf.continuous | case h.left.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheor... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | exact ContDiffBump.hasCompactSupport_normed φ | case h.left.hcf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousL... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hcf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheo... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | exact ContDiffBump.contDiff_normed φ | case h.left.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheor... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | refine Continuous.locallyIntegrable ?h.left.hg.hf | case h.left.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLi... | case h.left.hg.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (Continuou... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheor... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | exact unicontf.continuous | case h.left.hg.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (Continuou... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hg.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTh... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | intro x | case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLine... | case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLine... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | rw [← Complex.dist_eq, dist_comm] | case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLine... | case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLine... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | apply ContDiffBump.dist_normed_convolution_le | case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLine... | case h.right.hmg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (Continuous... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | . exact unicontf.continuous.aestronglyMeasurable | case h.right.hmg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (Continuous... | case h.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousL... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.hmg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureThe... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | . intro y hy
simp only [Metric.mem_ball] at hy
exact (hδ hy).le | case h.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousL... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheo... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | exact unicontf.continuous.aestronglyMeasurable | case h.right.hmg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (Continuous... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.hmg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureThe... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | intro y hy | case h.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousL... | case h.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousL... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheo... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | simp only [Metric.mem_ball] at hy | case h.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousL... | case h.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousL... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheo... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApprox | [43, 1] | [62, 23] | exact (hδ hy).le | case h.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousL... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f_0 : ℝ → ℂ :=
MeasureTheo... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | obtain ⟨δ, δpos, hδ⟩ := (Metric.uniformContinuous_iff.mp unicontf) ε εpos | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ Function.Periodic f₀ (2 * Real.pi) ∧ ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε | case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ Function.Periodic f₀ (2 * Real.pi) ∧ ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ Function.Periodic f₀ (2 * Real.pi) ∧ ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | let φ : ContDiffBump (0 : ℝ) := ⟨δ/2, δ, by linarith, by linarith⟩ | case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ Function.Periodic f₀ (2 * Real.pi) ∧ ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε | case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ Function.Periodi... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ Function.Periodic f₀ (2 * Real.pi) ∧ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | set f₀ := MeasureTheory.convolution (φ.normed MeasureTheory.volume) f (ContinuousLinearMap.lsmul ℝ ℝ) MeasureTheory.volume with f₀def | case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
⊢ ∃ f₀, ContDiff ℝ ⊤ f₀ ∧ Function.Periodi... | case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | use f₀ | case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution ... | case h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | constructor | case h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ.normed ... | case h.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ.no... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOu... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | .
apply HasCompactSupport.contDiff_convolution_left
. exact ContDiffBump.hasCompactSupport_normed φ
. exact ContDiffBump.contDiff_normed φ
. refine Continuous.locallyIntegrable ?h.left.hg.hf
exact unicontf.continuous | case h.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ.no... | case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ.n... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_l... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | constructor | case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ.n... | case h.right.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | .
intro x
rw [f₀def, MeasureTheory.convolution, MeasureTheory.convolution]
congr
ext t
congr 1
convert periodicf (x - t) using 2
ring | case h.right.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution... | case h.right.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolutio... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯,... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | . intro x
rw [← Complex.dist_eq, dist_comm]
apply ContDiffBump.dist_normed_convolution_le
. exact unicontf.continuous.aestronglyMeasurable
. intro y hy
simp only [Metric.mem_ball] at hy
exact (hδ hy).le | case h.right.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolutio... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | linarith | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ 0 < δ / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ 0 < δ / 2
TACTIC:
|
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