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/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply mul_pos hδ (by rw [C1_2]; norm_num) | case ha
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (part... | case hb
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (part... | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Real.rpow_pos_of_pos | case hb
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (part... | case hb.hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (p... | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | linarith [Real.two_pi_pos] | case hb.hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (p... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [C1_2] | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | norm_num | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [ε'def, C_control_approximation_effect_eq hε.1.le, add_sub_cancel_right, mul_div_cancel₀ _ Real.pi_pos.ne.symm] | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply mul_pos δ_mul_const_pos | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Real.rpow_pos_of_pos | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | case hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (part... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply div_pos (by norm_num) hε.1 | case hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (part... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | norm_num | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rwa [MeasureTheory.measureReal_def, MeasureTheory.measureReal_def, ←@ENNReal.toReal_ofReal 2 (by norm_num),
←ENNReal.toReal_mul, ENNReal.toReal_le_toReal Evolume.ne, ENNReal.ofReal_ofNat] | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply ENNReal.mul_ne_top ENNReal.ofReal_ne_top E'volume.ne | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | norm_num | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | conv => lhs; rw [←Real.rpow_one (MeasureTheory.volume.real E')] | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | congr | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | case e_a.e_a
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs ... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | norm_num | case e_a.e_a
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | gcongr | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | case h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (parti... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [Real.rpow_mul MeasureTheory.measureReal_nonneg] | case h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (parti... | case h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (parti... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | gcongr | case h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (parti... | case h.h₁
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (pa... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [Real.rpow_add' MeasureTheory.measureReal_nonneg (by norm_num), Real.rpow_one, le_div_iff' ε'_δ_expression_pos, ← mul_assoc] | case h.h₁
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (pa... | case h.h₁
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (pa... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ :=... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply mul_le_of_nonneg_of_le_div δ_mul_const_pos.le | case h.h₁
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (pa... | case h.h₁.hc
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ :=... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Real.rpow_nonneg MeasureTheory.measureReal_nonneg | case h.h₁.hc
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs ... | case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.hc
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw[Real.rpow_neg MeasureTheory.measureReal_nonneg, div_inv_eq_mul] | case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (... | case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [← ENNReal.ofReal_le_ofReal_iff, ENNReal.ofReal_mul ε'_δ_expression_pos.le, MeasureTheory.measureReal_def, ENNReal.ofReal_toReal E'volume.ne] | case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (... | case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply le_trans this | case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (... | case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [ENNReal.ofReal_mul δ_mul_const_pos.le, ← ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg (by norm_num), ENNReal.ofReal_toReal E'volume.ne] | case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (... | case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply mul_nonneg δ_mul_const_pos.le | case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (... | case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Real.rpow_nonneg MeasureTheory.measureReal_nonneg | case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | norm_num | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | norm_num | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [ε'def, C_control_approximation_effect_eq hε.1.le, add_sub_cancel_right, mul_div_cancel₀,
div_mul_eq_div_div, div_self, one_div, Real.inv_rpow, ← Real.rpow_mul, inv_mul_cancel, Real.rpow_one, inv_div] | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | ring | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | norm_num | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | case hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (part... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply div_nonneg <;> linarith [hε.1] | case hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (part... | case hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (part... | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Real.rpow_nonneg | case hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (part... | case hx.hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (p... | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply div_nonneg <;> linarith [hε.1] | case hx.hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (p... | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | Please generate a tactic in lean4 to solve the state.
STATE:
case hx.hx
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | exact δ_mul_const_pos.ne.symm | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFouri... | case hb
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (part... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | exact Real.pi_pos.ne.symm | case hb
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (part... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Proposition2.lean | prop2_2 | [36, 1] | [53, 8] | sorry | X : Type u_1
A : ℝ
inst✝⁶ : MetricSpace X
inst✝⁵ : IsSpaceOfHomogeneousType X A
inst✝⁴ : Inhabited X
τ q D κ ε C₀ C t : ℝ
Θ : Set C(X, ℂ)
inst✝³ : IsCompatible Θ
inst✝² : IsCancellative τ Θ
inst✝¹ : TileStructure Θ D κ C₀
F G : Set X
σ σ' : X → ℤ
Q' : X → C(X, ℂ)
K : X → X → ℂ
inst✝ : IsCZKernel τ K
ψ : ℝ → ℝ
𝔄 : Set ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
inst✝⁶ : MetricSpace X
inst✝⁵ : IsSpaceOfHomogeneousType X A
inst✝⁴ : Inhabited X
τ q D κ ε C₀ C t : ℝ
Θ : Set C(X, ℂ)
inst✝³ : IsCompatible Θ
inst✝² : IsCancellative τ Θ
inst✝¹ : TileStructure Θ D κ C₀
F G : Set X
σ σ' : X → ℤ
Q' : X → C(X... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_mul | [20, 1] | [28, 7] | simp only [fourierCoeffOn_eq_integral, one_div, fourier_apply, neg_smul, fourier_neg',
fourier_coe_apply', Complex.ofReal_sub, smul_eq_mul, Complex.real_smul, Complex.ofReal_inv] | a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
⊢ fourierCoeffOn hab (fun x => c * f x) n = c * fourierCoeffOn hab f n | a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
⊢ (↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (c * f x) =
c * ((↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
⊢ fourierCoeffOn hab (fun x => c * f x) n = c * fourierCoeffOn hab f n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_mul | [20, 1] | [28, 7] | rw [← mul_assoc, mul_comm c, mul_assoc _ c, mul_comm c, ← intervalIntegral.integral_mul_const] | a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
⊢ (↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (c * f x) =
c * ((↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x) | a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
⊢ (↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (c * f x) =
(↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x * c | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
⊢ (↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (c * f x) =
c * ((↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_mul | [20, 1] | [28, 7] | congr | a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
⊢ (↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (c * f x) =
(↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x * c | case e_a.e_f
a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
⊢ (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (c * f x)) = fun x =>
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x * c | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
⊢ (↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (c * f x) =
(↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_mul | [20, 1] | [28, 7] | ext x | case e_a.e_f
a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
⊢ (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (c * f x)) = fun x =>
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x * c | case e_a.e_f.h
a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
x : ℝ
⊢ (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (c * f x) =
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x * c | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
⊢ (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (c * f x)) = fun x =>
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x * c
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_mul | [20, 1] | [28, 7] | ring | case e_a.e_f.h
a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
x : ℝ
⊢ (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (c * f x) =
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
a b : ℝ
hab : a < b
f : ℝ → ℂ
c : ℂ
n : ℤ
x : ℝ
⊢ (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (c * f x) =
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x * c
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_neg | [30, 1] | [32, 98] | simp [fourierCoeffOn_eq_integral, fourierCoeffOn_eq_integral] | a b : ℝ
hab : a < b
f : ℝ → ℂ
n : ℤ
⊢ fourierCoeffOn hab (-f) n = -fourierCoeffOn hab f n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
f : ℝ → ℂ
n : ℤ
⊢ fourierCoeffOn hab (-f) n = -fourierCoeffOn hab f n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | simp only [fourierCoeffOn_eq_integral, one_div, fourier_apply, neg_smul, fourier_neg',
fourier_coe_apply', Complex.ofReal_sub, Pi.add_apply, smul_eq_mul, Complex.real_smul,
Complex.ofReal_inv] | a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ fourierCoeffOn hab (f + g) n = fourierCoeffOn hab f n + fourierCoeffOn hab g n | a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ (↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (f x + g x) =
((↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑... | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ fourierCoeffOn hab (f + g) n = fourierCoeffOn hab f n + fourierCoeffOn hab g n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | rw [← mul_add, ← intervalIntegral.integral_add] | a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ (↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (f x + g x) =
((↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑... | a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ (↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (f x + g x) =
(↑b - ↑a)⁻¹ *
∫ (x : ℝ) in a..b,
(starRing... | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ (↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (f x + g x) =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | congr | a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ (↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (f x + g x) =
(↑b - ↑a)⁻¹ *
∫ (x : ℝ) in a..b,
(starRing... | case e_a.e_f
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (f x + g x)) = fun x =>
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑... | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ (↑b - ↑a)⁻¹ * ∫ (x : ℝ) in a..b, (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (f x + g x) =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | ext x | case e_a.e_f
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (f x + g x)) = fun x =>
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑... | case e_a.e_f.h
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
x : ℝ
⊢ (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (f x + g x) =
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (f x + g x)) = fun x =>... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | ring | case e_a.e_f.h
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
x : ℝ
⊢ (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (f x + g x) =
(starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp... | case hf
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ IntervalIntegrable (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x)
MeasureTheory.volume a b
case hg
a b : ℝ
hab : a < b
f g : ℝ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
x : ℝ
⊢ (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * (f x + g x) =
(starRi... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | . apply hf.continuousOn_mul
apply Continuous.continuousOn
continuity | case hf
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ IntervalIntegrable (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x)
MeasureTheory.volume a b
case hg
a b : ℝ
hab : a < b
f g : ℝ... | case hg
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ IntervalIntegrable (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * g x)
MeasureTheory.volume a b | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ IntervalIntegrable (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x)
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | . apply hg.continuousOn_mul
apply Continuous.continuousOn
continuity | case hg
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ IntervalIntegrable (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * g x)
MeasureTheory.volume a b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ IntervalIntegrable (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * g x)
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | apply hf.continuousOn_mul | case hf
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ IntervalIntegrable (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x)
MeasureTheory.volume a b | case hf
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ ContinuousOn (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp) (Set.uIcc a b) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ IntervalIntegrable (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * f x)
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | apply Continuous.continuousOn | case hf
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ ContinuousOn (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp) (Set.uIcc a b) | case hf.h
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ Continuous fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ ContinuousOn (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp) (Set.uIcc a b)
T... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | continuity | case hf.h
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ Continuous fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.h
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ Continuous fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | apply hg.continuousOn_mul | case hg
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ IntervalIntegrable (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * g x)
MeasureTheory.volume a b | case hg
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ ContinuousOn (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp) (Set.uIcc a b) | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ IntervalIntegrable (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp * g x)
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | apply Continuous.continuousOn | case hg
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ ContinuousOn (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp) (Set.uIcc a b) | case hg.h
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ Continuous fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ ContinuousOn (fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp) (Set.uIcc a b)
T... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_add | [34, 1] | [49, 15] | continuity | case hg.h
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ Continuous fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg.h
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ Continuous fun x => (starRingEnd ℂ) (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (↑b - ↑a)).exp
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourierCoeffOn_sub | [51, 1] | [54, 90] | rw [sub_eq_add_neg, fourierCoeffOn_add hf hg.neg, fourierCoeffOn_neg, ← sub_eq_add_neg] | a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ fourierCoeffOn hab (f - g) n = fourierCoeffOn hab f n - fourierCoeffOn hab g n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
f g : ℝ → ℂ
n : ℤ
hf : IntervalIntegrable f MeasureTheory.volume a b
hg : IntervalIntegrable g MeasureTheory.volume a b
⊢ fourierCoeffOn hab (f - g) n = fourierCoeffOn hab f n - fourierCoeffOn hab g n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_add | [57, 1] | [63, 41] | ext x | f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
⊢ partialFourierSum (f + g) N = partialFourierSum f N + partialFourierSum g N | case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ partialFourierSum (f + g) N x = (partialFourierSum f N + partialFourierSum g N) x | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
⊢ partialFourierSum (f + g) N = partialFourierSum f N + partialFourierSum g N
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_add | [57, 1] | [63, 41] | simp [partialFourierSum, partialFourierSum, partialFourierSum, ← sum_add_distrib] | case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ partialFourierSum (f + g) N x = (partialFourierSum f N + partialFourierSum g N) x | case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ ∑ x_1 ∈ Icc (-↑N) ↑N,
fourierCoeffOn Real.two_pi_pos (f + g) x_1 * (2 * ↑Real.pi * Complex.I * ↑x_1 * ↑x / (2 * ↑Real.pi)).exp =
∑ x_1 ∈ Icc (-↑N) ↑N,
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ partialFourierSum (f + g) N x = (partialFourierSum f N + partialFourierSum g N) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_add | [57, 1] | [63, 41] | congr | case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ ∑ x_1 ∈ Icc (-↑N) ↑N,
fourierCoeffOn Real.two_pi_pos (f + g) x_1 * (2 * ↑Real.pi * Complex.I * ↑x_1 * ↑x / (2 * ↑Real.pi)).exp =
∑ x_1 ∈ Icc (-↑N) ↑N,
... | case h.e_f
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ (fun x_1 =>
fourierCoeffOn Real.two_pi_pos (f + g) x_1 * (2 * ↑Real.pi * Complex.I * ↑x_1 * ↑x / (2 * ↑Real.pi)).exp) =
fun x_1 =>
fourierCoeff... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ ∑ x_1 ∈ Icc (-↑N) ↑N,
fourierCoeffOn Real.two_pi_pos (f + g) x_1 * (2 * ↑Real.pi * Complex.I *... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_add | [57, 1] | [63, 41] | ext n | case h.e_f
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ (fun x_1 =>
fourierCoeffOn Real.two_pi_pos (f + g) x_1 * (2 * ↑Real.pi * Complex.I * ↑x_1 * ↑x / (2 * ↑Real.pi)).exp) =
fun x_1 =>
fourierCoeff... | case h.e_f.h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos (f + g) n * (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp =
fourierCoeffOn Real.two_pi_pos f n * (2 * ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_f
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ (fun x_1 =>
fourierCoeffOn Real.two_pi_pos (f + g) x_1 * (2 * ↑Real.pi * Complex.I * ↑x_1 ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_add | [57, 1] | [63, 41] | rw [fourierCoeffOn_add hf hg, add_mul] | case h.e_f.h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos (f + g) n * (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp =
fourierCoeffOn Real.two_pi_pos f n * (2 * ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_f.h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos (f + g) n * (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑R... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_sub | [65, 1] | [72, 41] | ext x | f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
⊢ partialFourierSum (f - g) N = partialFourierSum f N - partialFourierSum g N | case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ partialFourierSum (f - g) N x = (partialFourierSum f N - partialFourierSum g N) x | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
⊢ partialFourierSum (f - g) N = partialFourierSum f N - partialFourierSum g N
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_sub | [65, 1] | [72, 41] | simp only [Pi.sub_apply] | case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ partialFourierSum (f - g) N x = (partialFourierSum f N - partialFourierSum g N) x | case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ partialFourierSum (f - g) N x = partialFourierSum f N x - partialFourierSum g N x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ partialFourierSum (f - g) N x = (partialFourierSum f N - partialFourierSum g N) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_sub | [65, 1] | [72, 41] | rw [partialFourierSum, partialFourierSum, partialFourierSum, ←sum_sub_distrib] | case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ partialFourierSum (f - g) N x = partialFourierSum f N x - partialFourierSum g N x | case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos (f - g) n * (fourier n) ↑x =
∑ x_1 ∈ Icc (-Int.ofNat N) ↑N,
(fourierCoeffOn Real.two_pi_pos... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ partialFourierSum (f - g) N x = partialFourierSum f N x - partialFourierSum g N x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_sub | [65, 1] | [72, 41] | congr | case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos (f - g) n * (fourier n) ↑x =
∑ x_1 ∈ Icc (-Int.ofNat N) ↑N,
(fourierCoeffOn Real.two_pi_pos... | case h.e_f
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ (fun n => fourierCoeffOn Real.two_pi_pos (f - g) n * (fourier n) ↑x) = fun x_1 =>
fourierCoeffOn Real.two_pi_pos f x_1 * (fourier x_1) ↑x - fourierCoeffO... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos (f - g) n * (fourier n) ↑x =
∑ x_1 ∈... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_sub | [65, 1] | [72, 41] | ext n | case h.e_f
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ (fun n => fourierCoeffOn Real.two_pi_pos (f - g) n * (fourier n) ↑x) = fun x_1 =>
fourierCoeffOn Real.two_pi_pos f x_1 * (fourier x_1) ↑x - fourierCoeffO... | case h.e_f.h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos (f - g) n * (fourier n) ↑x =
fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x - fourierCoeffOn Real.two_pi_pos ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_f
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
⊢ (fun n => fourierCoeffOn Real.two_pi_pos (f - g) n * (fourier n) ↑x) = fun x_1 =>
fourierCoe... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_sub | [65, 1] | [72, 41] | rw [fourierCoeffOn_sub hf hg, sub_mul] | case h.e_f.h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos (f - g) n * (fourier n) ↑x =
fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x - fourierCoeffOn Real.two_pi_pos ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_f.h
f g : ℝ → ℂ
N : ℕ
hf : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
hg : IntervalIntegrable g MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos (f - g) n * (fourier n) ↑x =
fourierCoeffOn Real.two_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_mul | [75, 1] | [81, 37] | ext x | f : ℝ → ℂ
a : ℂ
N : ℕ
⊢ partialFourierSum (fun x => a * f x) N = fun x => a * partialFourierSum f N x | case h
f : ℝ → ℂ
a : ℂ
N : ℕ
x : ℝ
⊢ partialFourierSum (fun x => a * f x) N x = a * partialFourierSum f N x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
a : ℂ
N : ℕ
⊢ partialFourierSum (fun x => a * f x) N = fun x => a * partialFourierSum f N x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_mul | [75, 1] | [81, 37] | rw [partialFourierSum, partialFourierSum, mul_sum] | case h
f : ℝ → ℂ
a : ℂ
N : ℕ
x : ℝ
⊢ partialFourierSum (fun x => a * f x) N x = a * partialFourierSum f N x | case h
f : ℝ → ℂ
a : ℂ
N : ℕ
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos (fun x => a * f x) n * (fourier n) ↑x =
∑ i ∈ Icc (-Int.ofNat N) ↑N, a * (fourierCoeffOn Real.two_pi_pos f i * (fourier i) ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
a : ℂ
N : ℕ
x : ℝ
⊢ partialFourierSum (fun x => a * f x) N x = a * partialFourierSum f N x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_mul | [75, 1] | [81, 37] | congr | case h
f : ℝ → ℂ
a : ℂ
N : ℕ
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos (fun x => a * f x) n * (fourier n) ↑x =
∑ i ∈ Icc (-Int.ofNat N) ↑N, a * (fourierCoeffOn Real.two_pi_pos f i * (fourier i) ↑x) | case h.e_f
f : ℝ → ℂ
a : ℂ
N : ℕ
x : ℝ
⊢ (fun n => fourierCoeffOn Real.two_pi_pos (fun x => a * f x) n * (fourier n) ↑x) = fun i =>
a * (fourierCoeffOn Real.two_pi_pos f i * (fourier i) ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
a : ℂ
N : ℕ
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos (fun x => a * f x) n * (fourier n) ↑x =
∑ i ∈ Icc (-Int.ofNat N) ↑N, a * (fourierCoeffOn Real.two_pi_pos f i * (fourier i) ↑x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_mul | [75, 1] | [81, 37] | ext n | case h.e_f
f : ℝ → ℂ
a : ℂ
N : ℕ
x : ℝ
⊢ (fun n => fourierCoeffOn Real.two_pi_pos (fun x => a * f x) n * (fourier n) ↑x) = fun i =>
a * (fourierCoeffOn Real.two_pi_pos f i * (fourier i) ↑x) | case h.e_f.h
f : ℝ → ℂ
a : ℂ
N : ℕ
x : ℝ
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos (fun x => a * f x) n * (fourier n) ↑x =
a * (fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_f
f : ℝ → ℂ
a : ℂ
N : ℕ
x : ℝ
⊢ (fun n => fourierCoeffOn Real.two_pi_pos (fun x => a * f x) n * (fourier n) ↑x) = fun i =>
a * (fourierCoeffOn Real.two_pi_pos f i * (fourier i) ↑x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_mul | [75, 1] | [81, 37] | rw [fourierCoeffOn_mul, mul_assoc] | case h.e_f.h
f : ℝ → ℂ
a : ℂ
N : ℕ
x : ℝ
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos (fun x => a * f x) n * (fourier n) ↑x =
a * (fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_f.h
f : ℝ → ℂ
a : ℂ
N : ℕ
x : ℝ
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos (fun x => a * f x) n * (fourier n) ↑x =
a * (fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourier_periodic | [83, 1] | [83, 132] | simp | n : ℤ
⊢ Function.Periodic (fun x => (fourier n) ↑x) (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℤ
⊢ Function.Periodic (fun x => (fourier n) ↑x) (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_periodic | [85, 1] | [92, 27] | rw [Function.Periodic] | f : ℝ → ℂ
N : ℕ
⊢ Function.Periodic (partialFourierSum f N) (2 * Real.pi) | f : ℝ → ℂ
N : ℕ
⊢ ∀ (x : ℝ), partialFourierSum f N (x + 2 * Real.pi) = partialFourierSum f N x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
⊢ Function.Periodic (partialFourierSum f N) (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_periodic | [85, 1] | [92, 27] | intro x | f : ℝ → ℂ
N : ℕ
⊢ ∀ (x : ℝ), partialFourierSum f N (x + 2 * Real.pi) = partialFourierSum f N x | f : ℝ → ℂ
N : ℕ
x : ℝ
⊢ partialFourierSum f N (x + 2 * Real.pi) = partialFourierSum f N x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
⊢ ∀ (x : ℝ), partialFourierSum f N (x + 2 * Real.pi) = partialFourierSum f N x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_periodic | [85, 1] | [92, 27] | rw [partialFourierSum, partialFourierSum] | f : ℝ → ℂ
N : ℕ
x : ℝ
⊢ partialFourierSum f N (x + 2 * Real.pi) = partialFourierSum f N x | f : ℝ → ℂ
N : ℕ
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑(x + 2 * Real.pi) =
∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
x : ℝ
⊢ partialFourierSum f N (x + 2 * Real.pi) = partialFourierSum f N x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_periodic | [85, 1] | [92, 27] | congr | f : ℝ → ℂ
N : ℕ
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑(x + 2 * Real.pi) =
∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x | case e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
⊢ (fun n => fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑(x + 2 * Real.pi)) = fun n =>
fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑(x + 2 * Real.pi) =
∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_periodic | [85, 1] | [92, 27] | ext n | case e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
⊢ (fun n => fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑(x + 2 * Real.pi)) = fun n =>
fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x | case e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑(x + 2 * Real.pi) =
fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
⊢ (fun n => fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑(x + 2 * Real.pi)) = fun n =>
fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_periodic | [85, 1] | [92, 27] | congr 1 | case e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑(x + 2 * Real.pi) =
fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x | case e_f.h.e_a
f : ℝ → ℂ
N : ℕ
x : ℝ
n : ℤ
⊢ (fourier n) ↑(x + 2 * Real.pi) = (fourier n) ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑(x + 2 * Real.pi) =
fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_periodic | [85, 1] | [92, 27] | exact fourier_periodic x | case e_f.h.e_a
f : ℝ → ℂ
N : ℕ
x : ℝ
n : ℤ
⊢ (fourier n) ↑(x + 2 * Real.pi) = (fourier n) ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_a
f : ℝ → ℂ
N : ℕ
x : ℝ
n : ℤ
⊢ (fourier n) ↑(x + 2 * Real.pi) = (fourier n) ↑x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourier_uniformContinuous | [95, 1] | [98, 8] | simp [fourier] | n : ℤ
⊢ UniformContinuous fun x => (fourier n) ↑x | n : ℤ
⊢ UniformContinuous fun x => (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℤ
⊢ UniformContinuous fun x => (fourier n) ↑x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | fourier_uniformContinuous | [95, 1] | [98, 8] | sorry | n : ℤ
⊢ UniformContinuous fun x => (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℤ
⊢ UniformContinuous fun x => (2 * ↑Real.pi * Complex.I * ↑n * ↑x / (2 * ↑Real.pi)).exp
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | partialFourierSum_uniformContinuous | [100, 1] | [102, 8] | sorry | f : ℝ → ℂ
N : ℕ
⊢ UniformContinuous (partialFourierSum f N) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
⊢ UniformContinuous (partialFourierSum f N)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | strictConvexOn_cos_Icc | [105, 1] | [108, 55] | apply strictConvexOn_of_deriv2_pos (convex_Icc _ _) Real.continuousOn_cos fun x hx => ?_ | ⊢ StrictConvexOn ℝ (Set.Icc (Real.pi / 2) (Real.pi + Real.pi / 2)) Real.cos | x : ℝ
hx : x ∈ interior (Set.Icc (Real.pi / 2) (Real.pi + Real.pi / 2))
⊢ 0 < deriv^[2] Real.cos x | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ StrictConvexOn ℝ (Set.Icc (Real.pi / 2) (Real.pi + Real.pi / 2)) Real.cos
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | strictConvexOn_cos_Icc | [105, 1] | [108, 55] | rw [interior_Icc] at hx | x : ℝ
hx : x ∈ interior (Set.Icc (Real.pi / 2) (Real.pi + Real.pi / 2))
⊢ 0 < deriv^[2] Real.cos x | x : ℝ
hx : x ∈ Set.Ioo (Real.pi / 2) (Real.pi + Real.pi / 2)
⊢ 0 < deriv^[2] Real.cos x | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ interior (Set.Icc (Real.pi / 2) (Real.pi + Real.pi / 2))
⊢ 0 < deriv^[2] Real.cos x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | strictConvexOn_cos_Icc | [105, 1] | [108, 55] | simp [Real.cos_neg_of_pi_div_two_lt_of_lt hx.1 hx.2] | x : ℝ
hx : x ∈ Set.Ioo (Real.pi / 2) (Real.pi + Real.pi / 2)
⊢ 0 < deriv^[2] Real.cos x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Ioo (Real.pi / 2) (Real.pi + Real.pi / 2)
⊢ 0 < deriv^[2] Real.cos x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | by_cases ηpos : η ≤ 0 | η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | case pos
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : η ≤ 0
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
case neg
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : ¬η ≤ 0
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . calc (2 / Real.pi) * η
_ ≤ 0 := mul_nonpos_of_nonneg_of_nonpos (div_nonneg zero_le_two Real.pi_pos.le) ηpos
_ ≤ ‖1 - Complex.exp (Complex.I * x)‖ := by apply norm_nonneg | case pos
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : η ≤ 0
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
case neg
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : ¬η ≤ 0
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | case neg
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : ¬η ≤ 0
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : η ≤ 0
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
case neg
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : ¬η ≤ 0
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | push_neg at ηpos | case neg
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : ¬η ≤ 0
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | case neg
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : ¬η ≤ 0
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | wlog x_nonneg : 0 ≤ x generalizing x | case neg
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . convert (@this (-x) _ (by simpa) (by linarith)) using 1
. rw [Complex.norm_eq_abs, ←Complex.abs_conj, map_sub, map_one, Complex.ofReal_neg, mul_neg, Complex.norm_eq_abs,
←Complex.exp_conj, map_mul, Complex.conj_I, neg_mul, Complex.conj_ofReal]
. rwa [abs_neg] | case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 ... | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
η ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [abs_of_nonneg x_nonneg] at * | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | wlog x_le_pi : x ≤ Real.pi generalizing x | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | case inr
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x :... | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . convert (@this (2 * Real.pi - x) _ _ _ _) using 1
. rw [Complex.norm_eq_abs, ←Complex.abs_conj]
simp
rw [←Complex.exp_conj]
simp
rw [mul_sub, Complex.conj_ofReal, Complex.exp_sub, mul_comm Complex.I (2 * Real.pi), Complex.exp_two_pi_mul_I, ←inv_eq_one_div, ←Complex.exp_neg]
all_goals linarith | case inr
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x :... | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 / Real.pi * η ≤ ‖1... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | by_cases h : x ≤ Real.pi / 2 | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | case pos
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
case neg
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : ¬x ≤ R... | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . calc (2 / Real.pi) * η
_ ≤ (2 / Real.pi) * x := by gcongr
_ = (1 - (2 / Real.pi) * x) * Real.sin 0 + ((2 / Real.pi) * x) * Real.sin (Real.pi / 2) := by simp
_ ≤ Real.sin ((1 - (2 / Real.pi) * x) * 0 + ((2 / Real.pi) * x) * (Real.pi / 2)) := by
apply (strictConcaveOn_sin_Icc.concaveOn).2 (by simp [Real.pi_no... | case pos
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
case neg
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : ¬x ≤ R... | case neg
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : ¬x ≤ Real.pi / 2
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
case neg
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . push_neg at h
calc (2 / Real.pi) * η
_ ≤ (2 / Real.pi) * x := by gcongr
_ = 1 - ((1 - (2 / Real.pi) * (x - Real.pi / 2)) * Real.cos (Real.pi / 2) + ((2 / Real.pi) * (x - Real.pi / 2)) * Real.cos (Real.pi)) := by
field_simp
ring
_ ≤ 1 - (Real.cos ((1 - (2 / Real.pi) * (x - Real.pi / 2)) * (Real.pi / 2)... | case neg
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : ¬x ≤ Real.pi / 2
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : ¬x ≤ Real.pi / 2
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
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