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/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | have : (@Set.range α (z ∈ s) fun _ ↦ f z) = ∅ := by
simpa | case neg
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∉ s
⊢ y ≤ max x (sSup ∅) | case neg
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∉ s
this : (Set.range fun x => f z) = ∅
⊢ y ≤ max x (sSup ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∉ s
⊢ y ≤ max x (sSup ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | rw [this] at hz | case neg
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∉ s
this : (Set.range fun x => f z) = ∅
⊢ y ≤ max x (sSup ∅) | case neg
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup ∅ = y
h : z ∉ s
this : (Set.range fun x => f z) = ∅
⊢ y ≤ max x (sSup ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∉ s
this : (Set.range fun x => f z) = ∅
⊢ y ≤ max x (sSup ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | rw [hz] | case neg
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup ∅ = y
h : z ∉ s
this : (Set.range fun x => f z) = ∅
⊢ y ≤ max x (sSup ∅) | case neg
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup ∅ = y
h : z ∉ s
this : (Set.range fun x => f z) = ∅
⊢ y ≤ max x y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup ∅ = y
h : z ∉ s
this : (Set.range fun x => f z) = ∅
⊢ y ≤ max x (sSup ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | exact le_max_right x y | case neg
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup ∅ = y
h : z ∉ s
this : (Set.range fun x => f z) = ∅
⊢ y ≤ max x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup ∅ = y
h : z ∉ s
this : (Set.range fun x => f z) = ∅
⊢ y ≤ max x y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | have : (@Set.range α (z ∈ s) fun _ ↦ f z) = {f z} := by
rw [Set.eq_singleton_iff_unique_mem]
constructor
. simpa
. intro x hx
simp at hx
exact hx.2.symm | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ y ≤ max x (sSup ∅) | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
this : (Set.range fun x => f z) = {f z}
⊢ y ≤ max x (sSup ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ y ≤ max x (sSup ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | rw [this] at hz | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
this : (Set.range fun x => f z) = {f z}
⊢ y ≤ max x (sSup ∅) | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup {f z} = y
h : z ∈ s
this : (Set.range fun x => f z) = {f z}
⊢ y ≤ max x (sSup ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
this : (Set.range fun x => f z) = {f z}
⊢ y ≤ max x (sSup ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | have : sSup {f z} = f z := by apply csSup_singleton | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup {f z} = y
h : z ∈ s
this : (Set.range fun x => f z) = {f z}
⊢ y ≤ max x (sSup ∅) | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup {f z} = y
h : z ∈ s
this✝ : (Set.range fun x => f z) = {f z}
this : sSup {f z} = f z
⊢ y ≤ max x (sSup ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup {f z} = y
h : z ∈ s
this : (Set.range fun x => f z) = {f z}
⊢ y ≤ max x (sSup ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | rw [this] at hz | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup {f z} = y
h : z ∈ s
this✝ : (Set.range fun x => f z) = {f z}
this : sSup {f z} = f z
⊢ y ≤ max x (sSup ∅) | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : f z = y
h : z ∈ s
this✝ : (Set.range fun x => f z) = {f z}
this : sSup {f z} = f z
⊢ y ≤ max x (sSup ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup {f z} = y
h : z ∈ s
this✝ : (Set.range fun x => f z) = {f z}
this : sSup {f z} = f z
⊢ y ≤ max x (sSup ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | simp at hx | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : f z = y
h : z ∈ s
this✝ : (Set.range fun x => f z) = {f z}
this : sSup {f z} = f z
⊢ y ≤ max x (sSup ∅) | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x y : α
z : ι
hz : f z = y
h : z ∈ s
this✝ : (Set.range fun x => f z) = {f z}
this : sSup {f z} = f z
hx : ∀ a ∈ s, f a ≤ x
⊢ y ≤ max x (sSup ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : f z = y
h : z ∈ s
this✝ : (Set.range fun x => f z) = {f z}
this : sSup {f z} = f z
⊢ y ≤ max x (sSup ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | have : f z ≤ x := hx z h | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x y : α
z : ι
hz : f z = y
h : z ∈ s
this✝ : (Set.range fun x => f z) = {f z}
this : sSup {f z} = f z
hx : ∀ a ∈ s, f a ≤ x
⊢ y ≤ max x (sSup ∅) | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x y : α
z : ι
hz : f z = y
h : z ∈ s
this✝¹ : (Set.range fun x => f z) = {f z}
this✝ : sSup {f z} = f z
hx : ∀ a ∈ s, f a ≤ x
this : f z ≤ x
⊢ y ≤ max x (sSup ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x y : α
z : ι
hz : f z = y
h : z ∈ s
this✝ : (Set.range fun x => f z) = {f z}
this : sSup {f z} = f z
hx : ∀ a ∈ s, f a ≤ x
⊢ y ≤ max x (sSup ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | rw [hz] at this | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x y : α
z : ι
hz : f z = y
h : z ∈ s
this✝¹ : (Set.range fun x => f z) = {f z}
this✝ : sSup {f z} = f z
hx : ∀ a ∈ s, f a ≤ x
this : f z ≤ x
⊢ y ≤ max x (sSup ∅) | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x y : α
z : ι
hz : f z = y
h : z ∈ s
this✝¹ : (Set.range fun x => f z) = {f z}
this✝ : sSup {f z} = f z
hx : ∀ a ∈ s, f a ≤ x
this : y ≤ x
⊢ y ≤ max x (sSup ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x y : α
z : ι
hz : f z = y
h : z ∈ s
this✝¹ : (Set.range fun x => f z) = {f z}
this✝ : sSup {f z} = f z
hx : ∀ a ∈ s, f a ≤ x
this : f z ≤ x
⊢ y ≤ max x (sSup ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | apply le_max_of_le_left this | case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x y : α
z : ι
hz : f z = y
h : z ∈ s
this✝¹ : (Set.range fun x => f z) = {f z}
this✝ : sSup {f z} = f z
hx : ∀ a ∈ s, f a ≤ x
this : y ≤ x
⊢ y ≤ max x (sSup ∅) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x y : α
z : ι
hz : f z = y
h : z ∈ s
this✝¹ : (Set.range fun x => f z) = {f z}
this✝ : sSup {f z} = f z
hx : ∀ a ∈ s, f a ≤ x
this : y ≤ x
⊢ y ≤ max x (sSup ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | rw [Set.eq_singleton_iff_unique_mem] | α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ (Set.range fun x => f z) = {f z} | α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ (f z ∈ Set.range fun x => f z) ∧ ∀ x ∈ Set.range fun x => f z, x = f z | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ (Set.range fun x => f z) = {f z}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | constructor | α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ (f z ∈ Set.range fun x => f z) ∧ ∀ x ∈ Set.range fun x => f z, x = f z | case left
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ f z ∈ Set.range fun x => f z
case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ ∀ x ∈ Set.range fun x => f z, x = f z | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ (f z ∈ Set.range fun x => f z) ∧ ∀ x ∈ Set.range fun x => f z, x = f z
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | . simpa | case left
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ f z ∈ Set.range fun x => f z
case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ ∀ x ∈ Set.range fun x => f z, x = f z | case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ ∀ x ∈ Set.range fun x => f z, x = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ f z ∈ Set.range fun x => f z
case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ ∀ x ∈ Set.range fun x => f z, x = f z
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | . intro x hx
simp at hx
exact hx.2.symm | case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ ∀ x ∈ Set.range fun x => f z, x = f z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ ∀ x ∈ Set.range fun x => f z, x = f z
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | simpa | case left
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ f z ∈ Set.range fun x => f z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ f z ∈ Set.range fun x => f z
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | intro x hx | case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ ∀ x ∈ Set.range fun x => f z, x = f z | case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x✝ : α
hx✝ : ∀ y ∈ f '' s, y ≤ x✝
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
x : α
hx : x ∈ Set.range fun x => f z
⊢ x = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
⊢ ∀ x ∈ Set.range fun x => f z, x = f z
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | simp at hx | case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x✝ : α
hx✝ : ∀ y ∈ f '' s, y ≤ x✝
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
x : α
hx : x ∈ Set.range fun x => f z
⊢ x = f z | case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x✝ : α
hx✝ : ∀ y ∈ f '' s, y ≤ x✝
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
x : α
hx : z ∈ s ∧ f z = x
⊢ x = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x✝ : α
hx✝ : ∀ y ∈ f '' s, y ≤ x✝
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
x : α
hx : x ∈ Set.range fun x => f z
⊢ x = f z
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | exact hx.2.symm | case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x✝ : α
hx✝ : ∀ y ∈ f '' s, y ≤ x✝
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
x : α
hx : z ∈ s ∧ f z = x
⊢ x = f z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x✝ : α
hx✝ : ∀ y ∈ f '' s, y ≤ x✝
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∈ s
x : α
hx : z ∈ s ∧ f z = x
⊢ x = f z
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | apply csSup_singleton | α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup {f z} = y
h : z ∈ s
this : (Set.range fun x => f z) = {f z}
⊢ sSup {f z} = f z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup {f z} = y
h : z ∈ s
this : (Set.range fun x => f z) = {f z}
⊢ sSup {f z} = f z
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | simpa | α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∉ s
⊢ (Set.range fun x => f z) = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
ha : ∃ i ∈ s, f i = a
x : α
hx : ∀ y ∈ f '' s, y ≤ x
y : α
z : ι
hz : sSup (Set.range fun h => f z) = y
h : z ∉ s
⊢ (Set.range fun x => f z) = ∅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | simp | case h.e'_2.h.e'_3.left
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
⊢ a ∈ Set.range fun h => f i | case h.e'_2.h.e'_3.left
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
⊢ i ∈ s ∧ f i = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_3.left
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
⊢ a ∈ Set.range fun h => f i
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | use hi, fia | case h.e'_2.h.e'_3.left
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
⊢ i ∈ s ∧ f i = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_3.left
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
⊢ i ∈ s ∧ f i = a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | intro x hx | case h.e'_2.h.e'_3.right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
⊢ ∀ x ∈ Set.range fun h => f i, x = a | case h.e'_2.h.e'_3.right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
x : α
hx : x ∈ Set.range fun h => f i
⊢ x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_3.right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
⊢ ∀ x ∈ Set.range fun h => f i, x = a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | simp at hx | case h.e'_2.h.e'_3.right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
x : α
hx : x ∈ Set.range fun h => f i
⊢ x = a | case h.e'_2.h.e'_3.right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
x : α
hx : i ∈ s ∧ f i = x
⊢ x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_3.right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
x : α
hx : x ∈ Set.range fun h => f i
⊢ x = a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | ConditionallyCompleteLattice.le_biSup | [61, 1] | [104, 22] | rwa [hx.2] at fia | case h.e'_2.h.e'_3.right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
x : α
hx : i ∈ s ∧ f i = x
⊢ x = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_3.right
α : Type
inst✝¹ : ConditionallyCompleteLinearOrder α
ι : Type
inst✝ : Nonempty ι
f : ι → α
s : Set ι
a : α
hfs : BddAbove (f '' s)
i : ι
hi : i ∈ s
fia : f i = a
x : α
hx : i ∈ s ∧ f i = x
⊢ x = a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_same | [118, 1] | [120, 95] | rw [localOscillation] | X : Type
inst✝ : PseudoMetricSpace X
E : Set X
f : C(X, ℂ)
⊢ localOscillation E f f = 0 | X : Type
inst✝ : PseudoMetricSpace X
E : Set X
f : C(X, ℂ)
⊢ ⨆ z ∈ E ×ˢ E, ‖f z.1 - f z.1 - f z.2 + f z.2‖ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : PseudoMetricSpace X
E : Set X
f : C(X, ℂ)
⊢ localOscillation E f f = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_same | [118, 1] | [120, 95] | simp only [Set.mem_prod, sub_self, zero_sub, add_left_neg, norm_zero, Real.ciSup_const_zero] | X : Type
inst✝ : PseudoMetricSpace X
E : Set X
f : C(X, ℂ)
⊢ ⨆ z ∈ E ×ˢ E, ‖f z.1 - f z.1 - f z.2 + f z.2‖ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : PseudoMetricSpace X
E : Set X
f : C(X, ℂ)
⊢ ⨆ z ∈ E ×ˢ E, ‖f z.1 - f z.1 - f z.2 + f z.2‖ = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | intro n m | x R : ℝ
R_nonneg : 0 ≤ R
⊢ ∀ (n m : ℤ), localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
⊢ ∀ (n m : ℤ), localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | by_cases n_ne_m : n = m | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n = m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m|
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : ¬n = m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | . rw [n_ne_m]
simp
apply localOscillation_of_same | case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n = m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m|
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : ¬n = m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : ¬n = m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n = m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m|
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : ¬n = m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | push_neg at n_ne_m | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : ¬n = m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : ¬n = m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | have norm_n_sub_m_pos : 0 < |(n : ℝ) - m| := by
simp
rwa [sub_eq_zero, Int.cast_inj] | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | have norm_integer_linear_eq {n m : ℤ} {z : ℝ × ℝ} : ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ := by
rw [←Complex.norm_real, integer_linear, integer_linear]
congr 1
simp
ring | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | have localOscillation_eq : localOscillation (Metric.ball x R) (θ n) (θ m) = ⨆ z ∈ (Metric.ball x R) ×ˢ (Metric.ball x R), ‖(n - m) * (z.1 - x) - (n - m) * (z.2 - x)‖ := by
rw [localOscillation]
congr
ext z
rw [norm_integer_linear_eq] | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [localOscillation_eq] | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ = 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | apply le_antisymm | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ = 2 * R * |↑n - ↑m| | case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 2 * R * |↑n - ↑m| ≤ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ = 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | . calc ⨆ z ∈ (Metric.ball x R) ×ˢ (Metric.ball x R), ‖(n - m) * (z.1 - x) - (n - m) * (z.2 - x)‖
_ ≤ 2 * R * |↑n - ↑m| := by
apply Real.iSup_le
intro z
apply Real.iSup_le
. intro hz
simp at hz
rw [Real.dist_eq, Real.dist_eq] at hz
rw [Real.norm_eq_abs]
calc |(n - m) * (z.1 - x) - (n - m) * (z.2 - x)|
_ ≤ |(n - m) * (z.1 - x)| + |(n - m) * (z.2 - x)| := by apply abs_sub
_ = |↑n - ↑m| * |z.1 - x| + |↑n - ↑m| * |z.2 - x| := by congr <;> apply abs_mul
_ ≤ |↑n - ↑m| * R + |↑n - ↑m| * R := by gcongr; linarith [hz.1]; linarith [hz.2]
_ = 2 * R * |↑n - ↑m| := by ring
repeat
apply mul_nonneg
linarith
apply abs_nonneg | case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 2 * R * |↑n - ↑m| ≤ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 2 * R * |↑n - ↑m| ≤ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 2 * R * |↑n - ↑m| ≤ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | . apply le_of_forall_lt
intro c hc
have := fact_isCompact_ball x R
by_cases c_nonneg : 0 > c
. calc c
_ < 0 := c_nonneg
_ ≤ @dist (withLocalOscillation (Metric.ball x R)) PseudoMetricSpace.toDist (θ n) (θ m) := dist_nonneg
_ = localOscillation (Metric.ball x R) (θ n) (θ m) := rfl
_ = ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ := by
rw [localOscillation]
congr
ext z
congr
rw [norm_integer_linear_eq]
push_neg at c_nonneg
set R' := (c + 2 * R * |(n : ℝ) - m|) / (4 * |(n : ℝ) - m|) with R'def
have hR' : 0 ≤ R' ∧ R' < R := by
rw [R'def]
constructor
. positivity
calc (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
_ < (2 * R * |↑n - ↑m| + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) := by
gcongr
_ = R := by
ring_nf
rw [mul_assoc, mul_inv_cancel norm_n_sub_m_pos.ne.symm, mul_one]
let y := (x - R', x + R')
calc c
_ = c / 2 + c / 2 := by ring
_ < c / 2 + (2 * R * |↑n - ↑m|) / 2 := by
gcongr
_ = 2 * R' * |↑n - ↑m| := by
rw [R'def]
ring_nf
rw [pow_two, ←mul_assoc, mul_assoc c, mul_inv_cancel norm_n_sub_m_pos.ne.symm, mul_assoc (R * _), mul_inv_cancel norm_n_sub_m_pos.ne.symm]
ring
_ ≤ ‖(↑n - ↑m) * (y.1 - x) - (↑n - ↑m) * (y.2 - x)‖ := by
simp
rw [sub_eq_add_neg (-((n - m) * R')), ←neg_add, abs_neg, ←two_mul, abs_mul, abs_mul, mul_comm |(n : ℝ) - m|, mul_assoc]
simp
gcongr
apply le_abs_self
_ ≤ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ := by
apply ConditionallyCompleteLattice.le_biSup
. convert bddAbove_localOscillation (Metric.ball x R) (θ n) (θ m)
apply norm_integer_linear_eq.symm
. use y
simp
rw [abs_of_nonneg]
exact hR'.2
exact hR'.1 | case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 2 * R * |↑n - ↑m| ≤ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 2 * R * |↑n - ↑m| ≤ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [n_ne_m] | case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n = m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m| | case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n = m
⊢ localOscillation (Metric.ball x R) (θ m) (θ m) = 2 * R * |↑m - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n = m
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) = 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | simp | case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n = m
⊢ localOscillation (Metric.ball x R) (θ m) (θ m) = 2 * R * |↑m - ↑m| | case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n = m
⊢ localOscillation (Metric.ball x R) (θ m) (θ m) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n = m
⊢ localOscillation (Metric.ball x R) (θ m) (θ m) = 2 * R * |↑m - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | apply localOscillation_of_same | case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n = m
⊢ localOscillation (Metric.ball x R) (θ m) (θ m) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n = m
⊢ localOscillation (Metric.ball x R) (θ m) (θ m) = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | simp | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
⊢ 0 < |↑n - ↑m| | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
⊢ ¬↑n - ↑m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
⊢ 0 < |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rwa [sub_eq_zero, Int.cast_inj] | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
⊢ ¬↑n - ↑m = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
⊢ ¬↑n - ↑m = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [←Complex.norm_real, integer_linear, integer_linear] | x R : ℝ
R_nonneg : 0 ≤ R
n✝ m✝ : ℤ
n_ne_m : n✝ ≠ m✝
norm_n_sub_m_pos : 0 < |↑n✝ - ↑m✝|
n m : ℤ
z : ℝ × ℝ
⊢ ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | x R : ℝ
R_nonneg : 0 ≤ R
n✝ m✝ : ℤ
n_ne_m : n✝ ≠ m✝
norm_n_sub_m_pos : 0 < |↑n✝ - ↑m✝|
n m : ℤ
z : ℝ × ℝ
⊢ ‖{ toFun := fun x => ↑n * ↑x, continuous_toFun := ⋯ } z.1 - { toFun := fun x => ↑m * ↑x, continuous_toFun := ⋯ } z.1 -
{ toFun := fun x => ↑n * ↑x, continuous_toFun := ⋯ } z.2 +
{ toFun := fun x => ↑m * ↑x, continuous_toFun := ⋯ } z.2‖ =
‖↑((↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x))‖ | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n✝ m✝ : ℤ
n_ne_m : n✝ ≠ m✝
norm_n_sub_m_pos : 0 < |↑n✝ - ↑m✝|
n m : ℤ
z : ℝ × ℝ
⊢ ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | congr 1 | x R : ℝ
R_nonneg : 0 ≤ R
n✝ m✝ : ℤ
n_ne_m : n✝ ≠ m✝
norm_n_sub_m_pos : 0 < |↑n✝ - ↑m✝|
n m : ℤ
z : ℝ × ℝ
⊢ ‖{ toFun := fun x => ↑n * ↑x, continuous_toFun := ⋯ } z.1 - { toFun := fun x => ↑m * ↑x, continuous_toFun := ⋯ } z.1 -
{ toFun := fun x => ↑n * ↑x, continuous_toFun := ⋯ } z.2 +
{ toFun := fun x => ↑m * ↑x, continuous_toFun := ⋯ } z.2‖ =
‖↑((↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x))‖ | case e_a
x R : ℝ
R_nonneg : 0 ≤ R
n✝ m✝ : ℤ
n_ne_m : n✝ ≠ m✝
norm_n_sub_m_pos : 0 < |↑n✝ - ↑m✝|
n m : ℤ
z : ℝ × ℝ
⊢ { toFun := fun x => ↑n * ↑x, continuous_toFun := ⋯ } z.1 - { toFun := fun x => ↑m * ↑x, continuous_toFun := ⋯ } z.1 -
{ toFun := fun x => ↑n * ↑x, continuous_toFun := ⋯ } z.2 +
{ toFun := fun x => ↑m * ↑x, continuous_toFun := ⋯ } z.2 =
↑((↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)) | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n✝ m✝ : ℤ
n_ne_m : n✝ ≠ m✝
norm_n_sub_m_pos : 0 < |↑n✝ - ↑m✝|
n m : ℤ
z : ℝ × ℝ
⊢ ‖{ toFun := fun x => ↑n * ↑x, continuous_toFun := ⋯ } z.1 - { toFun := fun x => ↑m * ↑x, continuous_toFun := ⋯ } z.1 -
{ toFun := fun x => ↑n * ↑x, continuous_toFun := ⋯ } z.2 +
{ toFun := fun x => ↑m * ↑x, continuous_toFun := ⋯ } z.2‖ =
‖↑((↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x))‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | simp | case e_a
x R : ℝ
R_nonneg : 0 ≤ R
n✝ m✝ : ℤ
n_ne_m : n✝ ≠ m✝
norm_n_sub_m_pos : 0 < |↑n✝ - ↑m✝|
n m : ℤ
z : ℝ × ℝ
⊢ { toFun := fun x => ↑n * ↑x, continuous_toFun := ⋯ } z.1 - { toFun := fun x => ↑m * ↑x, continuous_toFun := ⋯ } z.1 -
{ toFun := fun x => ↑n * ↑x, continuous_toFun := ⋯ } z.2 +
{ toFun := fun x => ↑m * ↑x, continuous_toFun := ⋯ } z.2 =
↑((↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)) | case e_a
x R : ℝ
R_nonneg : 0 ≤ R
n✝ m✝ : ℤ
n_ne_m : n✝ ≠ m✝
norm_n_sub_m_pos : 0 < |↑n✝ - ↑m✝|
n m : ℤ
z : ℝ × ℝ
⊢ ↑n * ↑z.1 - ↑m * ↑z.1 - ↑n * ↑z.2 + ↑m * ↑z.2 = (↑n - ↑m) * (↑z.1 - ↑x) - (↑n - ↑m) * (↑z.2 - ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
x R : ℝ
R_nonneg : 0 ≤ R
n✝ m✝ : ℤ
n_ne_m : n✝ ≠ m✝
norm_n_sub_m_pos : 0 < |↑n✝ - ↑m✝|
n m : ℤ
z : ℝ × ℝ
⊢ { toFun := fun x => ↑n * ↑x, continuous_toFun := ⋯ } z.1 - { toFun := fun x => ↑m * ↑x, continuous_toFun := ⋯ } z.1 -
{ toFun := fun x => ↑n * ↑x, continuous_toFun := ⋯ } z.2 +
{ toFun := fun x => ↑m * ↑x, continuous_toFun := ⋯ } z.2 =
↑((↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | ring | case e_a
x R : ℝ
R_nonneg : 0 ≤ R
n✝ m✝ : ℤ
n_ne_m : n✝ ≠ m✝
norm_n_sub_m_pos : 0 < |↑n✝ - ↑m✝|
n m : ℤ
z : ℝ × ℝ
⊢ ↑n * ↑z.1 - ↑m * ↑z.1 - ↑n * ↑z.2 + ↑m * ↑z.2 = (↑n - ↑m) * (↑z.1 - ↑x) - (↑n - ↑m) * (↑z.2 - ↑x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
x R : ℝ
R_nonneg : 0 ≤ R
n✝ m✝ : ℤ
n_ne_m : n✝ ≠ m✝
norm_n_sub_m_pos : 0 < |↑n✝ - ↑m✝|
n m : ℤ
z : ℝ × ℝ
⊢ ↑n * ↑z.1 - ↑m * ↑z.1 - ↑n * ↑z.2 + ↑m * ↑z.2 = (↑n - ↑m) * (↑z.1 - ↑x) - (↑n - ↑m) * (↑z.2 - ↑x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [localOscillation] | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | congr | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case e_s
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ (fun z => ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖) = fun z =>
⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | ext z | case e_s
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ (fun z => ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖) = fun z =>
⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case e_s.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ =
⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_s
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ (fun z => ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖) = fun z =>
⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [norm_integer_linear_eq] | case e_s.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ =
⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_s.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ =
⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | calc ⨆ z ∈ (Metric.ball x R) ×ˢ (Metric.ball x R), ‖(n - m) * (z.1 - x) - (n - m) * (z.2 - x)‖
_ ≤ 2 * R * |↑n - ↑m| := by
apply Real.iSup_le
intro z
apply Real.iSup_le
. intro hz
simp at hz
rw [Real.dist_eq, Real.dist_eq] at hz
rw [Real.norm_eq_abs]
calc |(n - m) * (z.1 - x) - (n - m) * (z.2 - x)|
_ ≤ |(n - m) * (z.1 - x)| + |(n - m) * (z.2 - x)| := by apply abs_sub
_ = |↑n - ↑m| * |z.1 - x| + |↑n - ↑m| * |z.2 - x| := by congr <;> apply abs_mul
_ ≤ |↑n - ↑m| * R + |↑n - ↑m| * R := by gcongr; linarith [hz.1]; linarith [hz.2]
_ = 2 * R * |↑n - ↑m| := by ring
repeat
apply mul_nonneg
linarith
apply abs_nonneg | case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | apply Real.iSup_le | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m| | case hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ∀ (i : ℝ × ℝ),
⨆ (_ : i ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (i.1 - x) - (↑n - ↑m) * (i.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | intro z | case hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ∀ (i : ℝ × ℝ),
⨆ (_ : i ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (i.1 - x) - (↑n - ↑m) * (i.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m| | case hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ∀ (i : ℝ × ℝ),
⨆ (_ : i ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (i.1 - x) - (↑n - ↑m) * (i.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | apply Real.iSup_le | case hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m| | case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ z ∈ Metric.ball x R ×ˢ Metric.ball x R → ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
case hS.ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ 0 ≤ 2 * R * |↑n - ↑m|
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | . intro hz
simp at hz
rw [Real.dist_eq, Real.dist_eq] at hz
rw [Real.norm_eq_abs]
calc |(n - m) * (z.1 - x) - (n - m) * (z.2 - x)|
_ ≤ |(n - m) * (z.1 - x)| + |(n - m) * (z.2 - x)| := by apply abs_sub
_ = |↑n - ↑m| * |z.1 - x| + |↑n - ↑m| * |z.2 - x| := by congr <;> apply abs_mul
_ ≤ |↑n - ↑m| * R + |↑n - ↑m| * R := by gcongr; linarith [hz.1]; linarith [hz.2]
_ = 2 * R * |↑n - ↑m| := by ring | case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ z ∈ Metric.ball x R ×ˢ Metric.ball x R → ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
case hS.ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ 0 ≤ 2 * R * |↑n - ↑m|
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m| | case hS.ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ 0 ≤ 2 * R * |↑n - ↑m|
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ z ∈ Metric.ball x R ×ˢ Metric.ball x R → ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
case hS.ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ 0 ≤ 2 * R * |↑n - ↑m|
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | repeat
apply mul_nonneg
linarith
apply abs_nonneg | case hS.ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ 0 ≤ 2 * R * |↑n - ↑m|
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ 0 ≤ 2 * R * |↑n - ↑m|
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | intro hz | case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ z ∈ Metric.ball x R ×ˢ Metric.ball x R → ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m| | case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : z ∈ Metric.ball x R ×ˢ Metric.ball x R
⊢ ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
⊢ z ∈ Metric.ball x R ×ˢ Metric.ball x R → ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | simp at hz | case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : z ∈ Metric.ball x R ×ˢ Metric.ball x R
⊢ ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m| | case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : dist z.1 x < R ∧ dist z.2 x < R
⊢ ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : z ∈ Metric.ball x R ×ˢ Metric.ball x R
⊢ ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [Real.dist_eq, Real.dist_eq] at hz | case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : dist z.1 x < R ∧ dist z.2 x < R
⊢ ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m| | case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : dist z.1 x < R ∧ dist z.2 x < R
⊢ ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [Real.norm_eq_abs] | case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m| | case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)| ≤ 2 * R * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ ≤ 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | calc |(n - m) * (z.1 - x) - (n - m) * (z.2 - x)|
_ ≤ |(n - m) * (z.1 - x)| + |(n - m) * (z.2 - x)| := by apply abs_sub
_ = |↑n - ↑m| * |z.1 - x| + |↑n - ↑m| * |z.2 - x| := by congr <;> apply abs_mul
_ ≤ |↑n - ↑m| * R + |↑n - ↑m| * R := by gcongr; linarith [hz.1]; linarith [hz.2]
_ = 2 * R * |↑n - ↑m| := by ring | case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)| ≤ 2 * R * |↑n - ↑m| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hS.hS
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)| ≤ 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | apply abs_sub | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)| ≤ |(↑n - ↑m) * (z.1 - x)| + |(↑n - ↑m) * (z.2 - x)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)| ≤ |(↑n - ↑m) * (z.1 - x)| + |(↑n - ↑m) * (z.2 - x)|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | congr <;> apply abs_mul | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |(↑n - ↑m) * (z.1 - x)| + |(↑n - ↑m) * (z.2 - x)| = |↑n - ↑m| * |z.1 - x| + |↑n - ↑m| * |z.2 - x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |(↑n - ↑m) * (z.1 - x)| + |(↑n - ↑m) * (z.2 - x)| = |↑n - ↑m| * |z.1 - x| + |↑n - ↑m| * |z.2 - x|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | gcongr | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |↑n - ↑m| * |z.1 - x| + |↑n - ↑m| * |z.2 - x| ≤ |↑n - ↑m| * R + |↑n - ↑m| * R | case h₁.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |z.1 - x| ≤ R
case h₂.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |z.2 - x| ≤ R | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |↑n - ↑m| * |z.1 - x| + |↑n - ↑m| * |z.2 - x| ≤ |↑n - ↑m| * R + |↑n - ↑m| * R
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | linarith [hz.1] | case h₁.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |z.1 - x| ≤ R
case h₂.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |z.2 - x| ≤ R | case h₂.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |z.2 - x| ≤ R | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |z.1 - x| ≤ R
case h₂.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |z.2 - x| ≤ R
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | linarith [hz.2] | case h₂.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |z.2 - x| ≤ R | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |z.2 - x| ≤ R
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | ring | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |↑n - ↑m| * R + |↑n - ↑m| * R = 2 * R * |↑n - ↑m| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
z : ℝ × ℝ
hz : |z.1 - x| < R ∧ |z.2 - x| < R
⊢ |↑n - ↑m| * R + |↑n - ↑m| * R = 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | apply mul_nonneg | case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m| | case ha.ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R
case ha.hb
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R * |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | linarith | case ha.ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R
case ha.hb
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ |↑n - ↑m| | case ha.hb
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
case ha.ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ 2 * R
case ha.hb
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | apply abs_nonneg | case ha.hb
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ |↑n - ↑m| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha.hb
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 0 ≤ |↑n - ↑m|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | apply le_of_forall_lt | case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 2 * R * |↑n - ↑m| ≤ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case neg.a.H
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ∀ c < 2 * R * |↑n - ↑m|, c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ 2 * R * |↑n - ↑m| ≤ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | intro c hc | case neg.a.H
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ∀ c < 2 * R * |↑n - ↑m|, c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case neg.a.H
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.H
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
⊢ ∀ c < 2 * R * |↑n - ↑m|, c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | have := fact_isCompact_ball x R | case neg.a.H
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case neg.a.H
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.H
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | by_cases c_nonneg : 0 > c | case neg.a.H
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : ¬0 > c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.H
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | . calc c
_ < 0 := c_nonneg
_ ≤ @dist (withLocalOscillation (Metric.ball x R)) PseudoMetricSpace.toDist (θ n) (θ m) := dist_nonneg
_ = localOscillation (Metric.ball x R) (θ n) (θ m) := rfl
_ = ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ := by
rw [localOscillation]
congr
ext z
congr
rw [norm_integer_linear_eq] | case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : ¬0 > c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : ¬0 > c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : ¬0 > c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | push_neg at c_nonneg | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : ¬0 > c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : ¬0 > c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | set R' := (c + 2 * R * |(n : ℝ) - m|) / (4 * |(n : ℝ) - m|) with R'def | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | have hR' : 0 ≤ R' ∧ R' < R := by
rw [R'def]
constructor
. positivity
calc (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
_ < (2 * R * |↑n - ↑m| + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) := by
gcongr
_ = R := by
ring_nf
rw [mul_assoc, mul_inv_cancel norm_n_sub_m_pos.ne.symm, mul_one] | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | let y := (x - R', x + R') | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
y : ℝ × ℝ := (x - R', x + R')
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | calc c
_ = c / 2 + c / 2 := by ring
_ < c / 2 + (2 * R * |↑n - ↑m|) / 2 := by
gcongr
_ = 2 * R' * |↑n - ↑m| := by
rw [R'def]
ring_nf
rw [pow_two, ←mul_assoc, mul_assoc c, mul_inv_cancel norm_n_sub_m_pos.ne.symm, mul_assoc (R * _), mul_inv_cancel norm_n_sub_m_pos.ne.symm]
ring
_ ≤ ‖(↑n - ↑m) * (y.1 - x) - (↑n - ↑m) * (y.2 - x)‖ := by
simp
rw [sub_eq_add_neg (-((n - m) * R')), ←neg_add, abs_neg, ←two_mul, abs_mul, abs_mul, mul_comm |(n : ℝ) - m|, mul_assoc]
simp
gcongr
apply le_abs_self
_ ≤ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ := by
apply ConditionallyCompleteLattice.le_biSup
. convert bddAbove_localOscillation (Metric.ball x R) (θ n) (θ m)
apply norm_integer_linear_eq.symm
. use y
simp
rw [abs_of_nonneg]
exact hR'.2
exact hR'.1 | case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
y : ℝ × ℝ := (x - R', x + R')
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
y : ℝ × ℝ := (x - R', x + R')
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | calc c
_ < 0 := c_nonneg
_ ≤ @dist (withLocalOscillation (Metric.ball x R)) PseudoMetricSpace.toDist (θ n) (θ m) := dist_nonneg
_ = localOscillation (Metric.ball x R) (θ n) (θ m) := rfl
_ = ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ := by
rw [localOscillation]
congr
ext z
congr
rw [norm_integer_linear_eq] | case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ c < ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [localOscillation] | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | congr | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case e_s
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ (fun z => ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖) = fun z =>
⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ ⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | ext z | case e_s
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ (fun z => ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖) = fun z =>
⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case e_s.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
z : ℝ × ℝ
⊢ ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ =
⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_s
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
⊢ (fun z => ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖) = fun z =>
⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | congr | case e_s.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
z : ℝ × ℝ
⊢ ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ =
⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | case e_s.h.e_s
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
z : ℝ × ℝ
⊢ (fun h => ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖) = fun h => ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_s.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
z : ℝ × ℝ
⊢ ⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ =
⨆ (_ : z ∈ Metric.ball x R ×ˢ Metric.ball x R), ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [norm_integer_linear_eq] | case e_s.h.e_s
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
z : ℝ × ℝ
⊢ (fun h => ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖) = fun h => ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_s.h.e_s
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 > c
z : ℝ × ℝ
⊢ (fun h => ‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖) = fun h => ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [R'def] | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ 0 ≤ R' ∧ R' < R | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ 0 ≤ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) ∧ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) < R | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ 0 ≤ R' ∧ R' < R
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | constructor | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ 0 ≤ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) ∧ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) < R | case left
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ 0 ≤ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
case right
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) < R | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ 0 ≤ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) ∧ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) < R
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | . positivity | case left
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ 0 ≤ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
case right
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) < R | case right
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) < R | Please generate a tactic in lean4 to solve the state.
STATE:
case left
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ 0 ≤ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
case right
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) < R
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | calc (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
_ < (2 * R * |↑n - ↑m| + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) := by
gcongr
_ = R := by
ring_nf
rw [mul_assoc, mul_inv_cancel norm_n_sub_m_pos.ne.symm, mul_one] | case right
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) < R | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) < R
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | positivity | case left
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ 0 ≤ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ 0 ≤ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | gcongr | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) < (2 * R * |↑n - ↑m| + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) < (2 * R * |↑n - ↑m| + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | ring_nf | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ (2 * R * |↑n - ↑m| + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) = R | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ R * |↑n - ↑m| * |↑n - ↑m|⁻¹ = R | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ (2 * R * |↑n - ↑m| + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|) = R
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [mul_assoc, mul_inv_cancel norm_n_sub_m_pos.ne.symm, mul_one] | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ R * |↑n - ↑m| * |↑n - ↑m|⁻¹ = R | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
⊢ R * |↑n - ↑m| * |↑n - ↑m|⁻¹ = R
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | ring | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
y : ℝ × ℝ := (x - R', x + R')
⊢ c = c / 2 + c / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
y : ℝ × ℝ := (x - R', x + R')
⊢ c = c / 2 + c / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | gcongr | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
y : ℝ × ℝ := (x - R', x + R')
⊢ c / 2 + c / 2 < c / 2 + 2 * R * |↑n - ↑m| / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
y : ℝ × ℝ := (x - R', x + R')
⊢ c / 2 + c / 2 < c / 2 + 2 * R * |↑n - ↑m| / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [R'def] | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
y : ℝ × ℝ := (x - R', x + R')
⊢ c / 2 + 2 * R * |↑n - ↑m| / 2 = 2 * R' * |↑n - ↑m| | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
y : ℝ × ℝ := (x - R', x + R')
⊢ c / 2 + 2 * R * |↑n - ↑m| / 2 = 2 * ((c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)) * |↑n - ↑m| | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Metric.ball x R ×ˢ Metric.ball x R, ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
c : ℝ
hc : c < 2 * R * |↑n - ↑m|
this : Fact (Bornology.IsBounded (Metric.ball x R))
c_nonneg : 0 ≤ c
R' : ℝ := (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
R'def : R' = (c + 2 * R * |↑n - ↑m|) / (4 * |↑n - ↑m|)
hR' : 0 ≤ R' ∧ R' < R
y : ℝ × ℝ := (x - R', x + R')
⊢ c / 2 + 2 * R * |↑n - ↑m| / 2 = 2 * R' * |↑n - ↑m|
TACTIC:
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