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/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simp | case ha.hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 1 / |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha.hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 1 / |y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | apply div_nonneg <;> simp | case hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ |y - y'| / |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ |y - y'| / |y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | C1_2_pos | [11, 1] | [17, 18] | rw [C1_2] | a q : ℝ
hq : 1 < q
⊢ 0 < C1_2 a q | a q : ℝ
hq : 1 < q
⊢ 0 < 2 ^ (450 * a ^ 3) / (q - 1) ^ 5 | Please generate a tactic in lean4 to solve the state.
STATE:
a q : ℝ
hq : 1 < q
⊢ 0 < C1_2 a q
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | C1_2_pos | [11, 1] | [17, 18] | apply div_pos | a q : ℝ
hq : 1 < q
⊢ 0 < 2 ^ (450 * a ^ 3) / (q - 1) ^ 5 | case ha
a q : ℝ
hq : 1 < q
⊢ 0 < 2 ^ (450 * a ^ 3)
case hb
a q : ℝ
hq : 1 < q
⊢ 0 < (q - 1) ^ 5 | Please generate a tactic in lean4 to solve the state.
STATE:
a q : ℝ
hq : 1 < q
⊢ 0 < 2 ^ (450 * a ^ 3) / (q - 1) ^ 5
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | C1_2_pos | [11, 1] | [17, 18] | . apply Real.rpow_pos_of_pos
norm_num | case ha
a q : ℝ
hq : 1 < q
⊢ 0 < 2 ^ (450 * a ^ 3)
case hb
a q : ℝ
hq : 1 < q
⊢ 0 < (q - 1) ^ 5 | case hb
a q : ℝ
hq : 1 < q
⊢ 0 < (q - 1) ^ 5 | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
a q : ℝ
hq : 1 < q
⊢ 0 < 2 ^ (450 * a ^ 3)
case hb
a q : ℝ
hq : 1 < q
⊢ 0 < (q - 1) ^ 5
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | C1_2_pos | [11, 1] | [17, 18] | . apply pow_pos
linarith [hq] | case hb
a q : ℝ
hq : 1 < q
⊢ 0 < (q - 1) ^ 5 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
a q : ℝ
hq : 1 < q
⊢ 0 < (q - 1) ^ 5
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | C1_2_pos | [11, 1] | [17, 18] | apply Real.rpow_pos_of_pos | case ha
a q : ℝ
hq : 1 < q
⊢ 0 < 2 ^ (450 * a ^ 3) | case ha.hx
a q : ℝ
hq : 1 < q
⊢ 0 < 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
a q : ℝ
hq : 1 < q
⊢ 0 < 2 ^ (450 * a ^ 3)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | C1_2_pos | [11, 1] | [17, 18] | norm_num | case ha.hx
a q : ℝ
hq : 1 < q
⊢ 0 < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha.hx
a q : ℝ
hq : 1 < q
⊢ 0 < 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | C1_2_pos | [11, 1] | [17, 18] | apply pow_pos | case hb
a q : ℝ
hq : 1 < q
⊢ 0 < (q - 1) ^ 5 | case hb.H
a q : ℝ
hq : 1 < q
⊢ 0 < q - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
a q : ℝ
hq : 1 < q
⊢ 0 < (q - 1) ^ 5
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | C1_2_pos | [11, 1] | [17, 18] | linarith [hq] | case hb.H
a q : ℝ
hq : 1 < q
⊢ 0 < q - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.H
a q : ℝ
hq : 1 < q
⊢ 0 < q - 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | Int.floor_le_iff | [62, 1] | [63, 65] | rw_mod_cast [← Int.floor_le_sub_one_iff, add_sub_cancel_right] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | Int.le_ceil_iff | [65, 1] | [66, 58] | rw_mod_cast [← Int.add_one_le_ceil_iff, sub_add_cancel] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | mem_nonzeroS_iff | [68, 1] | [77, 34] | rw [Set.mem_Ioo, nonzeroS, Finset.mem_Icc] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | mem_nonzeroS_iff | [68, 1] | [77, 34] | simp only [Int.floor_le_iff, neg_add_rev, Int.le_ceil_iff, lt_add_neg_iff_add_lt, sub_add_cancel,
mul_inv_rev] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | mem_nonzeroS_iff | [68, 1] | [77, 34] | rw [← lt_div_iff hx, mul_comm D⁻¹, ← div_lt_div_iff hx (by positivity), ← Real.logb_inv,
← Real.logb_inv, div_inv_eq_mul, ← zpow_add_one₀ (by positivity)] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | mem_nonzeroS_iff | [68, 1] | [77, 34] | simp_rw [← Real.rpow_intCast] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | mem_nonzeroS_iff | [68, 1] | [77, 34] | rw [Real.lt_logb_iff_rpow_lt hD (by positivity), Real.logb_lt_iff_lt_rpow hD (by positivity)] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | mem_nonzeroS_iff | [68, 1] | [77, 34] | simp [div_eq_mul_inv, mul_comm] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | mem_nonzeroS_iff | [68, 1] | [77, 34] | positivity | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | mem_nonzeroS_iff | [68, 1] | [77, 34] | positivity | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | mem_nonzeroS_iff | [68, 1] | [77, 34] | positivity | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | mem_nonzeroS_iff | [68, 1] | [77, 34] | positivity | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | psi_ne_zero_iff | [79, 1] | [81, 50] | rw [← mem_support, h2ψ, mem_nonzeroS_iff hx hD] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | psi_eq_zero_iff | [83, 1] | [85, 51] | rw [← iff_not_comm, ← psi_ne_zero_iff h2ψ hx hD] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x✝ y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | sum_Ks | [90, 1] | [99, 34] | have h2 : 0 < dist x y := dist_pos.mpr h | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | sum_Ks | [90, 1] | [99, 34] | simp_rw [Ks, ← Finset.mul_sum] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | sum_Ks | [90, 1] | [99, 34] | norm_cast | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | sum_Ks | [90, 1] | [99, 34] | suffices ∑ i in s, ψ (D ^ i * dist x y) = 1 by
simp [this] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | sum_Ks | [90, 1] | [99, 34] | rw [← Finset.sum_subset hs, h3ψ _ h2] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | sum_Ks | [90, 1] | [99, 34] | intros | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | sum_Ks | [90, 1] | [99, 34] | rwa [psi_eq_zero_iff h2ψ h2 hD] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | sum_Ks | [90, 1] | [99, 34] | simp [this] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y : X
hD✝ : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s✝ : ℤ
x y :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | intro rhs | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | have h_rhs : 0 ≤ rhs := by
sorry | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | rw [CarlesonOperator] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | refine Real.iSup_le (fun Q ↦ ?_) h_rhs | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | refine Real.iSup_le (fun hQ ↦ ?_) h_rhs | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | refine Real.iSup_le (fun r ↦ ?_) h_rhs | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | refine Real.iSup_le (fun R ↦ ?_) h_rhs | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | refine Real.iSup_le (fun hrR ↦ ?_) h_rhs | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | let σ : ℤ := ⌊Real.logb D (2 * r)⌋ + 1 | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | let σ' : ℤ := ⌈Real.logb D (4 * R)⌉ | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | trans Ce3_1 a τ q * maximalFunction f x +
‖∑ s in Finset.Icc σ σ', ∫ y, Ks K D ψ s x y * f y * exp (I * (Q y - Q x))‖ | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | rw [← sub_le_iff_le_add] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | simp_rw [mul_sub, Complex.exp_sub, mul_div, integral_div, ← Finset.sum_div,
norm_div] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | have h1 : ‖cexp (I * Q x)‖ = 1 := by
lift Q x to ℝ using hΘ Q hQ x with qx
simp only [mul_comm I qx, norm_exp_ofReal_mul_I] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | rw [h1, div_one] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | apply (norm_sub_norm_le _ _).trans | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | rw [← integral_finset_sum] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | simp_rw [← Finset.sum_mul] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | have h3 :
∫ (y : X) in {y | dist x y ∈ Set.Ioo r R}, K x y * f y * cexp (I * Q y) =
∫ (y : X) in {y | dist x y ∈ Set.Ioo r R}, (∑ x_1 in Finset.Icc σ σ', Ks K D ψ x_1 x y) * f y * cexp (I * Q y) := by
sorry | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | rw [h3, ← neg_sub, ← integral_univ, ← integral_diff] | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | all_goals sorry | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | sorry | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | lift Q x to ℝ using hΘ Q hQ x with qx | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | case intro
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | simp only [mul_comm I qx, norm_exp_ofReal_mul_I] | case intro
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | sorry | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation3_1 | [109, 1] | [144, 18] | sorry | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | equation2_2 | [177, 1] | [185, 8] | sorry | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | theorem1_2C | [191, 1] | [199, 8] | sorry | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Carleson.lean | theorem1_2C' | [203, 1] | [211, 8] | sorry | X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : X
hD : D > D1_1 a τ q
hψ : LipschitzWith (Cψ1_1 a τ q) ψ
h2ψ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
a : ℝ
inst✝⁵ : MetricSpace X
inst✝⁴ : IsSpaceOfHomogeneousType X a
inst✝³ : Inhabited X
τ q q' C : ℝ
Θ : Set C(X, ℂ)
inst✝² : IsCompatible Θ
inst✝¹ : IsCancellative a Θ
F G : Set X
K : X → X → ℂ
inst✝ : IsCZKernel a K
D : ℝ
ψ : ℝ → ℝ
s : ℤ
x y : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_periodic | [18, 1] | [19, 8] | sorry | N : ℕ
⊢ Function.Periodic (dirichletKernel N) (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
⊢ Function.Periodic (dirichletKernel N) (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel'_periodic | [21, 1] | [22, 8] | sorry | N : ℕ
⊢ Function.Periodic (dirichletKernel' N) (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
⊢ Function.Periodic (dirichletKernel' N) (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel'_measurable | [24, 1] | [28, 43] | apply Measurable.add | N : ℕ
⊢ Measurable (dirichletKernel' N) | case hf
N : ℕ
⊢ Measurable fun a => cexp (I * ↑N * ↑a) / (1 - cexp (-I * ↑a))
case hg
N : ℕ
⊢ Measurable fun a => cexp (-I * ↑N * ↑a) / (1 - cexp (I * ↑a)) | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
⊢ Measurable (dirichletKernel' N)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel'_measurable | [24, 1] | [28, 43] | . apply Measurable.div <;> measurability | case hf
N : ℕ
⊢ Measurable fun a => cexp (I * ↑N * ↑a) / (1 - cexp (-I * ↑a))
case hg
N : ℕ
⊢ Measurable fun a => cexp (-I * ↑N * ↑a) / (1 - cexp (I * ↑a)) | case hg
N : ℕ
⊢ Measurable fun a => cexp (-I * ↑N * ↑a) / (1 - cexp (I * ↑a)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
N : ℕ
⊢ Measurable fun a => cexp (I * ↑N * ↑a) / (1 - cexp (-I * ↑a))
case hg
N : ℕ
⊢ Measurable fun a => cexp (-I * ↑N * ↑a) / (1 - cexp (I * ↑a))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel'_measurable | [24, 1] | [28, 43] | . apply Measurable.div <;> measurability | case hg
N : ℕ
⊢ Measurable fun a => cexp (-I * ↑N * ↑a) / (1 - cexp (I * ↑a)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
N : ℕ
⊢ Measurable fun a => cexp (-I * ↑N * ↑a) / (1 - cexp (I * ↑a))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel'_measurable | [24, 1] | [28, 43] | apply Measurable.div <;> measurability | case hf
N : ℕ
⊢ Measurable fun a => cexp (I * ↑N * ↑a) / (1 - cexp (-I * ↑a)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
N : ℕ
⊢ Measurable fun a => cexp (I * ↑N * ↑a) / (1 - cexp (-I * ↑a))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel'_measurable | [24, 1] | [28, 43] | apply Measurable.div <;> measurability | case hg
N : ℕ
⊢ Measurable fun a => cexp (-I * ↑N * ↑a) / (1 - cexp (I * ↑a)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
N : ℕ
⊢ Measurable fun a => cexp (-I * ↑N * ↑a) / (1 - cexp (I * ↑a))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | have : (cexp (1 / 2 * I * x) - cexp (-1 / 2 * I * x)) * dirichletKernel N x
= cexp ((N + 1 / 2) * I * x) - cexp (-(N + 1 / 2) * I * x) := by
calc (cexp (1 / 2 * I * x) - cexp (-1 / 2 * I * x)) * dirichletKernel N x
_ = ∑ n in Icc (-Int.ofNat N) ↑N, (cexp ((n + 1 / 2) * I * ↑x) - cexp ((n - 1 / 2) * I * ↑x)) :... | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ dirichletKernel N x = dirichletKernel' N x | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
⊢ dirichletKernel N x = dirichletKernel' N x | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ dirichletKernel N x = dirichletKernel' N x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | have h' : (cexp (1 / 2 * I * x) - cexp (-1 / 2 * I * x)) ≠ 0 := by
contrapose! h
rw [sub_eq_zero] at h
calc cexp (I * ↑x)
_ = cexp (1 / 2 * I * ↑x) * cexp (1 / 2 * I * ↑x) := by
rw [← exp_add]
congr
rw [mul_assoc, ← mul_add]
ring
_ = cexp (1 / 2 * I * ↑x) * cexp (-1 / 2 * I * ↑x) :... | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
⊢ dirichletKernel N x = dirichletKernel' N x | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ dirichletKernel N x = dirichletKernel' N x | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
⊢ dirichletKernel N x = dirichletKernel' N x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [dirichletKernel'] | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ dirichletKernel N x = dirichletKernel' N x | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ dirichletKernel N x = cexp (I * ↑N * ↑x) / (1 - cexp (-I * ↑x)) + cexp (-I * ↑N *... | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ dirichletKernel N x ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | apply mul_left_cancel₀ h' | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ dirichletKernel N x = cexp (I * ↑N * ↑x) / (1 - cexp (-I * ↑x)) + cexp (-I * ↑N *... | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
(cex... | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ dirichletKernel N x ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [this, mul_add, sub_eq_add_neg] | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
(cex... | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N + 1 / 2) * I * ↑x) + -cexp (-(↑N + 1 / 2) * I * ↑x) =
(cexp (1 / 2 ... | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ (cexp (1 / 2 * I * ↑... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N + 1 / 2) * I * ↑x) + -cexp (-(↑N + 1 / 2) * I * ↑x) =
(cexp (1 / 2 ... | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N + 1 / 2) * I * ↑x) =
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2... | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N + 1 / 2) *... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | . rw [mul_div]
apply eq_div_of_mul_eq
. contrapose! h
rwa [sub_eq_zero, neg_mul, exp_neg, eq_comm, inv_eq_one] at h
ring_nf
rw [← exp_add, ← exp_add, ← exp_add]
congr 2 <;> ring | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N + 1 / 2) * I * ↑x) =
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2... | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(↑N + 1 / 2) * I * ↑x) =
(cexp (1 / 2 * I * ↑x) - cexp (-1 /... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N +... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | . rw [mul_div]
apply eq_div_of_mul_eq
. contrapose! h
rwa [sub_eq_zero, eq_comm] at h
ring_nf
rw [← exp_add, ← exp_add, ← exp_add, neg_add_eq_sub]
congr 2 <;> ring | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(↑N + 1 / 2) * I * ↑x) =
(cexp (1 / 2 * I * ↑x) - cexp (-1 /... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(↑N... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | calc (cexp (1 / 2 * I * x) - cexp (-1 / 2 * I * x)) * dirichletKernel N x
_ = ∑ n in Icc (-Int.ofNat N) ↑N, (cexp ((n + 1 / 2) * I * ↑x) - cexp ((n - 1 / 2) * I * ↑x)) := by
rw [dirichletKernel, mul_sum]
congr
ext n
simp [sub_mul, ← exp_add, ← exp_add]
congr <;>
. ring_nf
congr 1
r... | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [dirichletKernel, mul_sum] | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
∑ n ∈ Icc (-Int.ofNat N) ↑N, (cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x)) | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ i ∈ Icc (-Int.ofNat N) ↑N, (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * (fourier i) ↑x =
∑ n ∈ Icc (-Int.ofNat N) ↑N, (cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x)) | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
∑ n ∈ Icc (-Int.ofNat N) ↑N, (cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ i ∈ Icc (-Int.ofNat N) ↑N, (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * (fourier i) ↑x =
∑ n ∈ Icc (-Int.ofNat N) ↑N, (cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x)) | case e_f
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (fun i => (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * (fourier i) ↑x) = fun n =>
cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ i ∈ Icc (-Int.ofNat N) ↑N, (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * (fourier i) ↑x =
∑ n ∈ Icc (-Int.ofNat N) ↑N, (cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | ext n | case e_f
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (fun i => (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * (fourier i) ↑x) = fun n =>
cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x) | case e_f.h
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * (fourier n) ↑x =
cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (fun i => (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * (fourier i) ↑x) = fun n =>
cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | simp [sub_mul, ← exp_add, ← exp_add] | case e_f.h
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * (fourier n) ↑x =
cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x) | case e_f.h
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ cexp (2⁻¹ * I * ↑x + 2 * ↑Real.pi * I * ↑n * ↑x / (2 * ↑Real.pi)) -
cexp (-1 / 2 * I * ↑x + 2 * ↑Real.pi * I * ↑n * ↑x / (2 * ↑Real.pi)) =
cexp ((↑n + 2⁻¹) * I * ↑x) - cexp (↑n * I * ↑x - 2⁻¹ * I * ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ (cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * (fourier n) ↑x =
cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr <;>
. ring_nf
congr 1
rw [mul_assoc, mul_assoc]
congr
rw [← mul_assoc, mul_comm, ← mul_assoc, inv_mul_cancel, one_mul]
norm_cast
exact Real.pi_pos.ne.symm | case e_f.h
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ cexp (2⁻¹ * I * ↑x + 2 * ↑Real.pi * I * ↑n * ↑x / (2 * ↑Real.pi)) -
cexp (-1 / 2 * I * ↑x + 2 * ↑Real.pi * I * ↑n * ↑x / (2 * ↑Real.pi)) =
cexp ((↑n + 2⁻¹) * I * ↑x) - cexp (↑n * I * ↑x - 2⁻¹ * I * ↑x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ cexp (2⁻¹ * I * ↑x + 2 * ↑Real.pi * I * ↑n * ↑x / (2 * ↑Real.pi)) -
cexp (-1 / 2 * I * ↑x + 2 * ↑Real.pi * I * ↑n * ↑x / (2 * ↑Real.pi)) =
cexp ((↑n + 2⁻¹) * I * ↑x) - cexp (↑n * I * ↑x - 2⁻¹ * I ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | ring_nf | case e_f.h.e_a.e_z
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ -1 / 2 * I * ↑x + 2 * ↑Real.pi * I * ↑n * ↑x / (2 * ↑Real.pi) = ↑n * I * ↑x - 2⁻¹ * I * ↑x | case e_f.h.e_a.e_z
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ I * ↑x * (-1 / 2) + I * ↑x * ↑Real.pi * ↑n * (↑Real.pi)⁻¹ = I * ↑x * (-1 / 2) + I * ↑x * ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_a.e_z
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ -1 / 2 * I * ↑x + 2 * ↑Real.pi * I * ↑n * ↑x / (2 * ↑Real.pi) = ↑n * I * ↑x - 2⁻¹ * I * ↑x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr 1 | case e_f.h.e_a.e_z
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ I * ↑x * (-1 / 2) + I * ↑x * ↑Real.pi * ↑n * (↑Real.pi)⁻¹ = I * ↑x * (-1 / 2) + I * ↑x * ↑n | case e_f.h.e_a.e_z.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ I * ↑x * ↑Real.pi * ↑n * (↑Real.pi)⁻¹ = I * ↑x * ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_a.e_z
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ I * ↑x * (-1 / 2) + I * ↑x * ↑Real.pi * ↑n * (↑Real.pi)⁻¹ = I * ↑x * (-1 / 2) + I * ↑x * ↑n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [mul_assoc, mul_assoc] | case e_f.h.e_a.e_z.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ I * ↑x * ↑Real.pi * ↑n * (↑Real.pi)⁻¹ = I * ↑x * ↑n | case e_f.h.e_a.e_z.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ I * ↑x * (↑Real.pi * (↑n * (↑Real.pi)⁻¹)) = I * ↑x * ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_a.e_z.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ I * ↑x * ↑Real.pi * ↑n * (↑Real.pi)⁻¹ = I * ↑x * ↑n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr | case e_f.h.e_a.e_z.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ I * ↑x * (↑Real.pi * (↑n * (↑Real.pi)⁻¹)) = I * ↑x * ↑n | case e_f.h.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ ↑Real.pi * (↑n * (↑Real.pi)⁻¹) = ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_a.e_z.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ I * ↑x * (↑Real.pi * (↑n * (↑Real.pi)⁻¹)) = I * ↑x * ↑n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [← mul_assoc, mul_comm, ← mul_assoc, inv_mul_cancel, one_mul] | case e_f.h.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ ↑Real.pi * (↑n * (↑Real.pi)⁻¹) = ↑n | case e_f.h.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ ↑Real.pi ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ ↑Real.pi * (↑n * (↑Real.pi)⁻¹) = ↑n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | norm_cast | case e_f.h.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ ↑Real.pi ≠ 0 | case e_f.h.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ ¬Real.pi = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ ↑Real.pi ≠ 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | exact Real.pi_pos.ne.symm | case e_f.h.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ ¬Real.pi = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ ¬Real.pi = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [sum_sub_distrib] | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, (cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x)) =
∑ n ∈ Icc (-Int.ofNat N) ↑N, cexp ((↑n + 1 / 2) * I * ↑x) -
∑ n ∈ Icc (-Int.ofNat N) ↑N, cexp ((↑n - 1 / 2) * I * ↑x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, (cexp ((↑n + 1 / 2) * I * ↑x) - cexp ((↑n - 1 / 2) * I * ↑x)) =
∑ n ∈ Icc (-Int.ofNat N) ↑N, cexp ((↑n + 1 / 2) * I * ↑x) -
∑ n ∈ Icc (-Int.ofNat N) ↑N, cexp ((↑n - 1 / 2) * I * ↑x)
TA... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [← sum_Ico_add_eq_sum_Icc, ← sum_Ioc_add_eq_sum_Icc, add_sub_add_comm] | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, cexp ((↑n + 1 / 2) * I * ↑x) -
∑ n ∈ Icc (-Int.ofNat N) ↑N, cexp ((↑n - 1 / 2) * I * ↑x) =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x) | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x) +
(cexp ((↑↑N + 1 / 2) * I * ↑x) - cexp ((↑(-Int.ofNat N) - 1 / 2) * I * ↑x)) =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)... | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, cexp ((↑n + 1 / 2) * I * ↑x) -
∑ n ∈ Icc (-Int.ofNat N) ↑N, cexp ((↑n - 1 / 2) * I * ↑x) =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | symm | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x) +
(cexp ((↑↑N + 1 / 2) * I * ↑x) - cexp ((↑(-Int.ofNat N) - 1 / 2) * I * ↑x)) =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)... | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x) =
∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x) +
(cexp ((↑↑N + 1 / 2) * I * ↑x) - cexp ((↑(-Int.ofNat N) - 1 / 2) * I * ↑x))... | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x) +
(cexp ((↑↑N + 1 / 2) * I * ↑x) - cexp ((↑(-Int.ofNat N) - 1 / 2) * I * ↑x)) =
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [← zero_add (cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x))] | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x) =
∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x) +
(cexp ((↑↑N + 1 / 2) * I * ↑x) - cexp ((↑(-Int.ofNat N) - 1 / 2) * I * ↑x))... | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ 0 + (cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)) =
∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x) +
(cexp ((↑↑N + 1 / 2) * I * ↑x) - cexp ((↑(-Int.ofNat N) - 1 / 2) * I ... | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x) =
∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x) +
(cexp ((↑↑N + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ 0 + (cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)) =
∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x) +
(cexp ((↑↑N + 1 / 2) * I * ↑x) - cexp ((↑(-Int.ofNat N) - 1 / 2) * I ... | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ 0 =
∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ ... | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ 0 + (cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)) =
∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x) +
(cexp ((... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | symm | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ 0 =
∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ ... | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x) =
0
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ 0 =
∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [sub_eq_zero] | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x) =
0
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ ... | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) =
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.o... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) -
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x) =
0
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | conv => lhs; rw [← Int.add_sub_cancel (-Int.ofNat N) 1, sub_eq_add_neg, ← Int.add_sub_cancel (Nat.cast N) 1, sub_eq_add_neg, ← sum_Ico_add'] | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) =
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.o... | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ x_1 ∈ Ico (-Int.ofNat N + 1) (↑N + 1), cexp ((↑(x_1 + -1) + 1 / 2) * I * ↑x) =
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I *... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ x_1 ∈ Ico (-Int.ofNat N) ↑N, cexp ((↑x_1 + 1 / 2) * I * ↑x) =
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ x_1 ∈ Ico (-Int.ofNat N + 1) (↑N + 1), cexp ((↑(x_1 + -1) + 1 / 2) * I * ↑x) =
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I *... | case e_a.e_s
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ Ico (-Int.ofNat N + 1) (↑N + 1) = Ioc (-Int.ofNat N) ↑N
case e_a.e_f
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (fun x_1 => cexp ((↑(x_1 + -1) + 1 / 2) * I * ↑x)) = fun x_1 => cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ∑ x_1 ∈ Ico (-Int.ofNat N + 1) (↑N + 1), cexp ((↑(x_1 + -1) + 1 / 2) * I * ↑x) =
∑ x_1 ∈ Ioc (-Int.ofNat N) ↑N, cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | . ext n
rw [mem_Ico, mem_Ioc, Int.lt_iff_add_one_le, add_le_add_iff_right, ← mem_Icc, Int.lt_iff_add_one_le, ← mem_Icc] | case e_a.e_s
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ Ico (-Int.ofNat N + 1) (↑N + 1) = Ioc (-Int.ofNat N) ↑N
case e_a.e_f
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (fun x_1 => cexp ((↑(x_1 + -1) + 1 / 2) * I * ↑x)) = fun x_1 => cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1... | case e_a.e_f
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (fun x_1 => cexp ((↑(x_1 + -1) + 1 / 2) * I * ↑x)) = fun x_1 => cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N
N : ℕ
x : ℝ
h : ce... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_s
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ Ico (-Int.ofNat N + 1) (↑N + 1) = Ioc (-Int.ofNat N) ↑N
case e_a.e_f
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (fun x_1 => cexp ((↑(x_1 + -1) + 1 / 2) * I * ↑x)) = fun x_1 => cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | . ext n
congr
simp [add_assoc, sub_eq_add_neg]
congr
norm_num | case e_a.e_f
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (fun x_1 => cexp ((↑(x_1 + -1) + 1 / 2) * I * ↑x)) = fun x_1 => cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N
N : ℕ
x : ℝ
h : ce... | case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (fun x_1 => cexp ((↑(x_1 + -1) + 1 / 2) * I * ↑x)) = fun x_1 => cexp ((↑x_1 - 1 / 2) * I * ↑x)
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
... |
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