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/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | linarith | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ δ / 2 < δ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
⊢ δ / 2 < δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | apply HasCompactSupport.contDiff_convolution_left | case h.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ.no... | case h.left.hcf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_l... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | . exact ContDiffBump.hasCompactSupport_normed φ | case h.left.hcf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (... | case h.left.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hcf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, r... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | . exact ContDiffBump.contDiff_normed φ | case h.left.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ... | case h.left.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rI... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | . refine Continuous.locallyIntegrable ?h.left.hg.hf
exact unicontf.continuous | case h.left.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rI... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | exact ContDiffBump.hasCompactSupport_normed φ | case h.left.hcf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hcf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, r... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | exact ContDiffBump.contDiff_normed φ | case h.left.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rI... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | refine Continuous.locallyIntegrable ?h.left.hg.hf | case h.left.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution (φ... | case h.left.hg.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rI... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | exact unicontf.continuous | case h.left.hg.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hg.hf
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯,... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | intro x | case h.right.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution... | case h.right.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯,... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | rw [f₀def, MeasureTheory.convolution, MeasureTheory.convolution] | case h.right.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution... | case h.right.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯,... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | congr | case h.right.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolution... | case h.right.left.e_f
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯,... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | ext t | case h.right.left.e_f
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolu... | case h.right.left.e_f.h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convo... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left.e_f
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | congr 1 | case h.right.left.e_f.h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convo... | case h.right.left.e_f.h.h.e_6.h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheo... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left.e_f.h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | convert periodicf (x - t) using 2 | case h.right.left.e_f.h.h.e_6.h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheo... | case h.e'_2.h.e'_1
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolutio... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left.e_f.h.h.e_6.h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ,... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | ring | case h.e'_2.h.e'_1
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolutio... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_1
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | intro x | case h.right.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolutio... | case h.right.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolutio... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | rw [← Complex.dist_eq, dist_comm] | case h.right.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolutio... | case h.right.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolutio... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | apply ContDiffBump.dist_normed_convolution_le | case h.right.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolutio... | case h.right.right.hmg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convol... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | . exact unicontf.continuous.aestronglyMeasurable | case h.right.right.hmg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convol... | case h.right.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right.hmg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | . intro y hy
simp only [Metric.mem_ball] at hy
exact (hδ hy).le | case h.right.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | exact unicontf.continuous.aestronglyMeasurable | case h.right.right.hmg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convol... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right.hmg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | intro y hy | case h.right.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolu... | case h.right.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | simp only [Metric.mem_ball] at hy | case h.right.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolu... | case h.right.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | closeSmoothApproxPeriodic | [65, 1] | [93, 23] | exact (hδ hy).le | case h.right.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f₀ : ℝ → ℂ :=
MeasureTheory.convolu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right.hg
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
hδ : ∀ {a b : ℝ}, dist a b < δ → dist (f a) (f b) < ε
φ : ContDiffBump 0 := { rIn := δ / 2, rOut := δ, rIn_pos :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | Real.summable_of_le | [96, 1] | [98, 53] | exact Summable.of_nonneg_of_le hgpos hgf summablef | β : Type
f g : β → ℝ
hgpos : 0 ≤ g
hgf : ∀ (b : β), g b ≤ f b
summablef : Summable f
⊢ Summable g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
f g : β → ℝ
hgpos : 0 ≤ g
hgf : ∀ (b : β), g b ≤ f b
summablef : Summable f
⊢ Summable g
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | set f' : ℤ → ℝ := fun i ↦ if i = 0 then g i else f i with f'def | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
⊢ Summable g | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
⊢ Summable g | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
⊢ Summable g
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | have : ∀ i, g i ≤ f' i := by
intro i
rw [f'def]
by_cases h : i = 0
. simp [h]
. simp only [h, ↓reduceIte]
exact hgf i h | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
⊢ Summable g | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
⊢ Summable g | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
⊢ Summable g
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | apply Real.summable_of_le hgpos this | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
⊢ Summable g | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
⊢ Summable fun b => f' b | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
⊢ Summable g
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | let s : Finset ℤ := {0} | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
⊢ Summable fun b => f' b | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
⊢ Summable fun b => f' b | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
⊢ Summable fun b => f' b
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | rw [←s.summable_compl_iff] | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
⊢ Summable fun b => f' b | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
⊢ Summable fun x => f' ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
⊢ Summable fun b => f' b
TA... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | apply (summable_congr _).mpr (s.summable_compl_iff.mpr summablef) | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
⊢ Summable fun x => f' ↑x | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
⊢ ∀ (b : { x // x ∉ s }), f' ↑b = f ↑b | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
⊢ Summable fun x => f' ↑x
T... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | intro ⟨b, hb⟩ | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
⊢ ∀ (b : { x // x ∉ s }), f' ↑b = f ↑b | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
b : ℤ
hb : b ∉ s
⊢ f' ↑⟨b, hb⟩ = f ↑⟨b, hb⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
⊢ ∀ (b : { x // x ∉ s }), f... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | simp | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
b : ℤ
hb : b ∉ s
⊢ f' ↑⟨b, hb⟩ = f ↑⟨b, hb⟩ | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
b : ℤ
hb : b ∉ s
⊢ f' b = f b | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
b : ℤ
hb : b ∉ s
⊢ f' ↑⟨b, ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | rw [mem_singleton] at hb | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
b : ℤ
hb : b ∉ s
⊢ f' b = f b | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
b : ℤ
hb : ¬b = 0
⊢ f' b = f b | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
b : ℤ
hb : b ∉ s
⊢ f' b = f... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | rw [f'def] | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
b : ℤ
hb : ¬b = 0
⊢ f' b = f b | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
b : ℤ
hb : ¬b = 0
⊢ (fun i => if i = 0 then g i else f i) b = f b | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
b : ℤ
hb : ¬b = 0
⊢ f' b = ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | simp [hb] | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
b : ℤ
hb : ¬b = 0
⊢ (fun i => if i = 0 then g i else f i) b = f b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
this : ∀ (i : ℤ), g i ≤ f' i
s : Finset ℤ := {0}
b : ℤ
hb : ¬b = 0
⊢ (fun i ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | intro i | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
⊢ ∀ (i : ℤ), g i ≤ f' i | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
⊢ g i ≤ f' i | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
⊢ ∀ (i : ℤ), g i ≤ f' i
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | rw [f'def] | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
⊢ g i ≤ f' i | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
⊢ g i ≤ f' i
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | by_cases h : i = 0 | f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i | case pos
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : i = 0
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i
case neg
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i... | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | . simp [h] | case pos
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : i = 0
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i
case neg
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i... | case neg
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : ¬i = 0
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : i = 0
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i
ca... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | . simp only [h, ↓reduceIte]
exact hgf i h | case neg
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : ¬i = 0
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : ¬i = 0
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i
TA... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | simp [h] | case pos
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : i = 0
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : i = 0
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i
TAC... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | simp only [h, ↓reduceIte] | case neg
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : ¬i = 0
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i | case neg
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : ¬i = 0
⊢ g i ≤ f i | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : ¬i = 0
⊢ g i ≤ (fun i => if i = 0 then g i else f i) i
TA... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | summable_of_le_on_nonzero | [115, 1] | [133, 12] | exact hgf i h | case neg
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : ¬i = 0
⊢ g i ≤ f i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f g : ℤ → ℝ
hgpos : 0 ≤ g
hgf : ∀ (i : ℤ), i ≠ 0 → g i ≤ f i
summablef : Summable f
f' : ℤ → ℝ := fun i => if i = 0 then g i else f i
f'def : f' = fun i => if i = 0 then g i else f i
i : ℤ
h : ¬i = 0
⊢ g i ≤ f i
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | continuous_bounded | [135, 3] | [144, 18] | have interval_compact := (@isCompact_Icc ℝ _ _ _ 0 (2 * Real.pi)) | f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
⊢ ∃ C, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C | f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
⊢ ∃ C, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
⊢ ∃ C, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | continuous_bounded | [135, 3] | [144, 18] | have abs_f_continuousOn := Complex.continuous_abs.comp_continuousOn hf | f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
⊢ ∃ C, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C | f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
⊢ ∃ C, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
⊢ ∃ C, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | continuous_bounded | [135, 3] | [144, 18] | obtain ⟨a, _, ha⟩ := interval_compact.exists_isMaxOn (Set.nonempty_Icc.mpr Real.two_pi_pos.le) abs_f_continuousOn | f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
⊢ ∃ C, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C | case intro.intro
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi)) a
⊢ ∃ C, ∀ x ... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
⊢ ∃ C, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | continuous_bounded | [135, 3] | [144, 18] | set C := Complex.abs (f a) with C_def | case intro.intro
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi)) a
⊢ ∃ C, ∀ x ... | case intro.intro
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi)) a
C : ℝ := Co... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMax... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | continuous_bounded | [135, 3] | [144, 18] | use C | case intro.intro
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi)) a
C : ℝ := Co... | case h
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi)) a
C : ℝ := Complex.abs ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMax... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | continuous_bounded | [135, 3] | [144, 18] | intro x hx | case h
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi)) a
C : ℝ := Complex.abs ... | case h
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi)) a
C : ℝ := Complex.abs ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | continuous_bounded | [135, 3] | [144, 18] | rw [C_def] | case h
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi)) a
C : ℝ := Complex.abs ... | case h
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi)) a
C : ℝ := Complex.abs ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | continuous_bounded | [135, 3] | [144, 18] | rw [isMaxOn_iff] at ha | case h
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi)) a
C : ℝ := Complex.abs ... | case h
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : ∀ x ∈ Set.Icc 0 (2 * Real.pi), (⇑Complex.abs ∘ f) x ≤ (⇑Complex.abs ∘ f) a
C... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : IsMaxOn (⇑Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | continuous_bounded | [135, 3] | [144, 18] | exact ha x hx | case h
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : ∀ x ∈ Set.Icc 0 (2 * Real.pi), (⇑Complex.abs ∘ f) x ≤ (⇑Complex.abs ∘ f) a
C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
hf : ContinuousOn f (Set.Icc 0 (2 * Real.pi))
interval_compact : IsCompact (Set.Icc 0 (2 * Real.pi))
abs_f_continuousOn : ContinuousOn (⇑Complex.abs ∘ f) (Set.Icc 0 (2 * Real.pi))
a : ℝ
left✝ : a ∈ Set.Icc 0 (2 * Real.pi)
ha : ∀ x ∈ Set.Icc 0... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | obtain ⟨C, f_bounded⟩ := continuous_bounded f_continuous.continuousOn | f : ℝ → ℂ
f_continuous : Continuous f
⊢ ∃ C, ∀ (n : ℤ), Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C | case intro
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
⊢ ∃ C, ∀ (n : ℤ), Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
f_continuous : Continuous f
⊢ ∃ C, ∀ (n : ℤ), Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | use C | case intro
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
⊢ ∃ C, ∀ (n : ℤ), Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
⊢ ∀ (n : ℤ), Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
⊢ ∃ C, ∀ (n : ℤ), Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | intro n | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
⊢ ∀ (n : ℤ), Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
⊢ ∀ (n : ℤ), Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | rw [fourierCoeffOn_eq_integral] | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ Complex.abs ((1 / (2 * Real.pi - 0)) • ∫ (x : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑x • f x) ≤ C | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | simp only [sub_zero, one_div, mul_inv_rev, fourier_apply, neg_smul, fourier_neg',
fourier_coe_apply', Complex.ofReal_mul, Complex.ofReal_ofNat, smul_eq_mul, Complex.real_smul,
Complex.ofReal_inv, map_mul, map_inv₀, Complex.abs_ofReal, Complex.abs_ofNat] | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ Complex.abs ((1 / (2 * Real.pi - 0)) • ∫ (x : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑x • f x) ≤ C | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ |Real.pi|⁻¹ * 2⁻¹ *
Complex.abs
(∫ (x : ℝ) in 0 ..2 * Real.pi,
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x) ≤
C | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ Complex.abs ((1 / (2 * Real.pi - 0)) • ∫ (x : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑x • f x) ≤ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | field_simp | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ |Real.pi|⁻¹ * 2⁻¹ *
Complex.abs
(∫ (x : ℝ) in 0 ..2 * Real.pi,
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x) ≤
C | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ Complex.abs
(∫ (x : ℝ) in 0 ..2 * Real.pi,
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x) /
(|Real.pi| * 2) ≤
C | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ |Real.pi|⁻¹ * 2⁻¹ *
Complex.abs
(∫ (x : ℝ) in 0 ..2 * Real.pi,
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n *... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | rw [abs_of_nonneg Real.pi_pos.le, mul_comm Real.pi, div_le_iff Real.two_pi_pos, ←Complex.norm_eq_abs] | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ Complex.abs
(∫ (x : ℝ) in 0 ..2 * Real.pi,
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x) /
(|Real.pi| * 2) ≤
C | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ‖∫ (x : ℝ) in 0 ..2 * Real.pi, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x‖ ≤
C * (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ Complex.abs
(∫ (x : ℝ) in 0 ..2 * Real.pi,
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | calc ‖∫ (x : ℝ) in (0 : ℝ)..(2 * Real.pi), (starRingEnd ℂ) (Complex.exp (2 * Real.pi * Complex.I * n * x / (2 * Real.pi))) * f x‖
_ = ‖∫ (x : ℝ) in (0 : ℝ)..(2 * Real.pi), (starRingEnd ℂ) (Complex.exp (Complex.I * n * x)) * f x‖ := by
congr
ext x
congr
ring_nf
rw [mul_comm, ←mul_assoc, ←mul_assoc,... | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ‖∫ (x : ℝ) in 0 ..2 * Real.pi, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x‖ ≤
C * (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ‖∫ (x : ℝ) in 0 ..2 * Real.pi, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x‖ ≤
C * (2 * Real.pi)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | congr | f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ‖∫ (x : ℝ) in 0 ..2 * Real.pi, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x‖ =
‖∫ (x : ℝ) in 0 ..2 * Real.pi, (starRingEnd ℂ) (Complex.I * ↑n * ↑x).exp * f x‖ | case e_a.e_f
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x) = fun x =>
(starRingEnd ℂ) (Complex.I * ↑n * ↑x).exp * f x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ‖∫ (x : ℝ) in 0 ..2 * Real.pi, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x‖ =
‖∫ (x : ℝ) in 0 ..2 * Re... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | ext x | case e_a.e_f
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x) = fun x =>
(starRingEnd ℂ) (Complex.I * ↑n * ↑x).exp * f x | case e_a.e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x =
(starRingEnd ℂ) (Complex.I * ↑n * ↑x).exp * f x | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x) = fun x =>
(starRingEnd ℂ) (Comple... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | congr | case e_a.e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x =
(starRingEnd ℂ) (Complex.I * ↑n * ↑x).exp * f x | case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ 2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi) = Complex.I * ↑n * ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp * f x =
(starRingEnd ℂ) (Complex.I * ↑n * ↑... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | ring_nf | case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ 2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi) = Complex.I * ↑n * ↑x | case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ↑Real.pi * Complex.I * ↑n * ↑x * (↑Real.pi)⁻¹ = Complex.I * ↑n * ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ 2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi) = Complex.I * ↑n * ↑x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | rw [mul_comm, ←mul_assoc, ←mul_assoc, ←mul_assoc, inv_mul_cancel] | case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ↑Real.pi * Complex.I * ↑n * ↑x * (↑Real.pi)⁻¹ = Complex.I * ↑n * ↑x | case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ 1 * Complex.I * ↑n * ↑x = Complex.I * ↑n * ↑x
case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), C... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ↑Real.pi * Complex.I * ↑n * ↑x * (↑Real.pi)⁻¹ = Complex.I * ↑n * ↑x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | . ring | case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ 1 * Complex.I * ↑n * ↑x = Complex.I * ↑n * ↑x
case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), C... | case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ↑Real.pi ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ 1 * Complex.I * ↑n * ↑x = Complex.I * ↑n * ↑x
case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Co... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | . simp
exact Real.pi_pos.ne.symm | case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ↑Real.pi ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ↑Real.pi ≠ 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | ring | case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ 1 * Complex.I * ↑n * ↑x = Complex.I * ↑n * ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ 1 * Complex.I * ↑n * ↑x = Complex.I * ↑n * ↑x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | simp | case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ↑Real.pi ≠ 0 | case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ¬Real.pi = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ↑Real.pi ≠ 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | exact Real.pi_pos.ne.symm | case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ¬Real.pi = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_a.h.e_6.h.e_z
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ¬Real.pi = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | apply intervalIntegral.norm_integral_le_integral_norm Real.two_pi_pos.le | f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ‖∫ (x : ℝ) in 0 ..2 * Real.pi, (starRingEnd ℂ) (Complex.I * ↑n * ↑x).exp * f x‖ ≤
∫ (x : ℝ) in 0 ..2 * Real.pi, ‖(starRingEnd ℂ) (Complex.I * ↑n * ↑x).exp * f x‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ‖∫ (x : ℝ) in 0 ..2 * Real.pi, (starRingEnd ℂ) (Complex.I * ↑n * ↑x).exp * f x‖ ≤
∫ (x : ℝ) in 0 ..2 * Real.pi, ‖(starRingEnd ℂ) (Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | simp | f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∫ (x : ℝ) in 0 ..2 * Real.pi, ‖(starRingEnd ℂ) (Complex.I * ↑n * ↑x).exp * f x‖ =
∫ (x : ℝ) in 0 ..2 * Real.pi, ‖(Complex.I * ↑n * ↑x).exp * f x‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∫ (x : ℝ) in 0 ..2 * Real.pi, ‖(starRingEnd ℂ) (Complex.I * ↑n * ↑x).exp * f x‖ =
∫ (x : ℝ) in 0 ..2 * Real.pi, ‖(Complex.I * ↑n * ↑x).exp... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | congr | f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∫ (x : ℝ) in 0 ..2 * Real.pi, ‖(Complex.I * ↑n * ↑x).exp * f x‖ = ∫ (x : ℝ) in 0 ..2 * Real.pi, ‖f x‖ | case e_f
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ (fun x => ‖(Complex.I * ↑n * ↑x).exp * f x‖) = fun x => ‖f x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∫ (x : ℝ) in 0 ..2 * Real.pi, ‖(Complex.I * ↑n * ↑x).exp * f x‖ = ∫ (x : ℝ) in 0 ..2 * Real.pi, ‖f x‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | ext x | case e_f
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ (fun x => ‖(Complex.I * ↑n * ↑x).exp * f x‖) = fun x => ‖f x‖ | case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ‖(Complex.I * ↑n * ↑x).exp * f x‖ = ‖f x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ (fun x => ‖(Complex.I * ↑n * ↑x).exp * f x‖) = fun x => ‖f x‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | simp only [norm_mul, Complex.norm_eq_abs] | case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ‖(Complex.I * ↑n * ↑x).exp * f x‖ = ‖f x‖ | case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ Complex.abs (Complex.I * ↑n * ↑x).exp * Complex.abs (f x) = Complex.abs (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ ‖(Complex.I * ↑n * ↑x).exp * f x‖ = ‖f x‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | rw [mul_assoc, mul_comm Complex.I] | case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ Complex.abs (Complex.I * ↑n * ↑x).exp * Complex.abs (f x) = Complex.abs (f x) | case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ Complex.abs (↑n * ↑x * Complex.I).exp * Complex.abs (f x) = Complex.abs (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ Complex.abs (Complex.I * ↑n * ↑x).exp * Complex.abs (f x) = Complex.abs (f x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | norm_cast | case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ Complex.abs (↑n * ↑x * Complex.I).exp * Complex.abs (f x) = Complex.abs (f x) | case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ Complex.abs (↑(↑n * x) * Complex.I).exp * Complex.abs (f x) = Complex.abs (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ Complex.abs (↑n * ↑x * Complex.I).exp * Complex.abs (f x) = Complex.abs (f x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | rw [Complex.abs_exp_ofReal_mul_I] | case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ Complex.abs (↑(↑n * x) * Complex.I).exp * Complex.abs (f x) = Complex.abs (f x) | case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ 1 * Complex.abs (f x) = Complex.abs (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ Complex.abs (↑(↑n * x) * Complex.I).exp * Complex.abs (f x) = Complex.abs (f x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | ring | case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ 1 * Complex.abs (f x) = Complex.abs (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
⊢ 1 * Complex.abs (f x) = Complex.abs (f x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | apply intervalIntegral.integral_mono_on | f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∫ (x : ℝ) in 0 ..2 * Real.pi, ‖f x‖ ≤ ∫ (x : ℝ) in 0 ..2 * Real.pi, C | case hab
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ 0 ≤ 2 * Real.pi
case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => ‖f u‖) MeasureTheory.vo... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∫ (x : ℝ) in 0 ..2 * Real.pi, ‖f x‖ ≤ ∫ (x : ℝ) in 0 ..2 * Real.pi, C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | . exact Real.two_pi_pos.le | case hab
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ 0 ≤ 2 * Real.pi
case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => ‖f u‖) MeasureTheory.vo... | case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => ‖f u‖) MeasureTheory.volume 0 (2 * Real.pi)
case hg
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hab
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ 0 ≤ 2 * Real.pi
case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | . rw [IntervalIntegrable.intervalIntegrable_norm_iff]
. apply f_continuous.intervalIntegrable
. apply f_continuous.aestronglyMeasurable | case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => ‖f u‖) MeasureTheory.volume 0 (2 * Real.pi)
case hg
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ... | case hg
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => C) MeasureTheory.volume 0 (2 * Real.pi)
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∀... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => ‖f u‖) MeasureTheory.volume 0 (2 * Real.pi)
case hg
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounde... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | . apply intervalIntegrable_const | case hg
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => C) MeasureTheory.volume 0 (2 * Real.pi)
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∀... | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi), ‖f x‖ ≤ C | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => C) MeasureTheory.volume 0 (2 * Real.pi)
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | . intro x hx
rw [Complex.norm_eq_abs]
exact f_bounded x hx | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi), ‖f x‖ ≤ C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi), ‖f x‖ ≤ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | exact Real.two_pi_pos.le | case hab
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ 0 ≤ 2 * Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hab
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ 0 ≤ 2 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | rw [IntervalIntegrable.intervalIntegrable_norm_iff] | case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => ‖f u‖) MeasureTheory.volume 0 (2 * Real.pi) | case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => f u) MeasureTheory.volume 0 (2 * Real.pi)
case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => ‖f u‖) MeasureTheory.volume 0 (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | . apply f_continuous.intervalIntegrable | case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => f u) MeasureTheory.volume 0 (2 * Real.pi)
case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
... | case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ MeasureTheory.AEStronglyMeasurable (fun u => f u) (MeasureTheory.volume.restrict (Ι 0 (2 * Real.pi))) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => f u) MeasureTheory.volume 0 (2 * Real.pi)
case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | . apply f_continuous.aestronglyMeasurable | case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ MeasureTheory.AEStronglyMeasurable (fun u => f u) (MeasureTheory.volume.restrict (Ι 0 (2 * Real.pi))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ MeasureTheory.AEStronglyMeasurable (fun u => f u) (MeasureTheory.volume.restrict (Ι 0 (2 * Real.pi)))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | apply f_continuous.intervalIntegrable | case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => f u) MeasureTheory.volume 0 (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => f u) MeasureTheory.volume 0 (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | apply f_continuous.aestronglyMeasurable | case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ MeasureTheory.AEStronglyMeasurable (fun u => f u) (MeasureTheory.volume.restrict (Ι 0 (2 * Real.pi))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ MeasureTheory.AEStronglyMeasurable (fun u => f u) (MeasureTheory.volume.restrict (Ι 0 (2 * Real.pi)))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | apply intervalIntegrable_const | case hg
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => C) MeasureTheory.volume 0 (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ IntervalIntegrable (fun u => C) MeasureTheory.volume 0 (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | intro x hx | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi), ‖f x‖ ≤ C | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
⊢ ‖f x‖ ≤ C | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi), ‖f x‖ ≤ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | rw [Complex.norm_eq_abs] | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
⊢ ‖f x‖ ≤ C | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
⊢ Complex.abs (f x) ≤ C | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
⊢ ‖f x‖ ≤ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | exact f_bounded x hx | case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
⊢ Complex.abs (f x) ≤ C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
⊢ Complex.abs (f x) ≤ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | rw [intervalIntegral.integral_const] | f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∫ (x : ℝ) in 0 ..2 * Real.pi, C = C * (2 * Real.pi) | f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ (2 * Real.pi - 0) • C = C * (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ ∫ (x : ℝ) in 0 ..2 * Real.pi, C = C * (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | simp | f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ (2 * Real.pi - 0) • C = C * (2 * Real.pi) | f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ 2 * Real.pi * C = C * (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ (2 * Real.pi - 0) • C = C * (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_bound | [147, 1] | [193, 11] | ring | f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ 2 * Real.pi * C = C * (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
f_continuous : Continuous f
C : ℝ
f_bounded : ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x) ≤ C
n : ℤ
⊢ 2 * Real.pi * C = C * (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | intro x | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
⊢ Function.Periodic (deriv f) T | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
⊢ deriv f (x + T) = deriv f x | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
⊢ Function.Periodic (deriv f) T
TACTIC:
|
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