Split (1)

train · 219k rows

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/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	simp [hdef]	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	Please generate a tactic in lean4 to solve the state. STATE: f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	rw [←Complex.dist_eq, dist_comm, Complex.dist_eq]	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	Please generate a tactic in lean4 to solve the state. STATE: f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	exact hf₀ x	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	congr	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	case h.e_6.h f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Func...	Please generate a tactic in lean4 to solve the state. STATE: f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	ring	case h.e_6.h f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Func...	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.e_6.h f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	apply AbsoluteValue.add_le	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	apply add_le_add_right	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	case bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function....	Please generate a tactic in lean4 to solve the state. STATE: f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	apply AbsoluteValue.add_le	case bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function....	no goals	Please generate a tactic in lean4 to solve the state. STATE: case bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	gcongr	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	case h₁.h₁ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	Please generate a tactic in lean4 to solve the state. STATE: f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	. exact hf₀ x	case h₁.h₁ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	case h₁.h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	Please generate a tactic in lean4 to solve the state. STATE: case h₁.h₁ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	. exact hN₀ N NgtN₀ x hx.1	case h₁.h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function....	Please generate a tactic in lean4 to solve the state. STATE: case h₁.h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	. have := hE x hx N rw [hdef, partialFourierSum_sub (contDiff_f₀.continuous.intervalIntegrable 0 (2 * Real.pi)) (unicontf.continuous.intervalIntegrable 0 (2 * Real.pi))] at this apply le_trans this rw [ε'def, mul_div_cancel₀ _ (C_control_approximation_effect_pos εpos).ne.symm]	case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function....	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	exact hf₀ x	case h₁.h₁ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h₁.h₁ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	exact hN₀ N NgtN₀ x hx.1	case h₁.h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h₁.h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	have := hE x hx N	case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function....	case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function....	Please generate a tactic in lean4 to solve the state. STATE: case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	rw [hdef, partialFourierSum_sub (contDiff_f₀.continuous.intervalIntegrable 0 (2 * Real.pi)) (unicontf.continuous.intervalIntegrable 0 (2 * Real.pi))] at this	case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function....	case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function....	Please generate a tactic in lean4 to solve the state. STATE: case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	apply le_trans this	case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function....	case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function....	Please generate a tactic in lean4 to solve the state. STATE: case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	rw [ε'def, mul_div_cancel₀ _ (C_control_approximation_effect_pos εpos).ne.symm]	case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function....	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h₂ f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	gcongr	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	case bc.bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	Please generate a tactic in lean4 to solve the state. STATE: f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	rw [ε'def, div_div]	case bc.bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	case bc.bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	Please generate a tactic in lean4 to solve the state. STATE: case bc.bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	apply div_le_div_of_nonneg_left εpos.le (by norm_num)	case bc.bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	case bc.bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	Please generate a tactic in lean4 to solve the state. STATE: case bc.bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	rw [← div_le_iff' (by norm_num)]	case bc.bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	case bc.bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	Please generate a tactic in lean4 to solve the state. STATE: case bc.bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	apply le_trans' (lt_C_control_approximation_effect εpos).le (by linarith [Real.two_le_pi])	case bc.bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Functi...	no goals	Please generate a tactic in lean4 to solve the state. STATE: case bc.bc f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	norm_num	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	norm_num	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	linarith [Real.two_le_pi]	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/ClassicalCarleson.lean	classical_carleson	[13, 1]	[67, 23]	linarith	f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ contDiff_f₀ : ContDiff ℝ ⊤ f₀ periodic_f₀ : Function.Periodic...	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℝ → ℂ unicontf : UniformContinuous f periodicf : Function.Periodic f (2 * Real.pi) ε : ℝ εpos : 0 < ε εle : ε ≤ 2 * Real.pi ε' : ℝ := ε / 4 / C_control_approximation_effect ε ε'def : ε' = ε / 4 / C_control_approximation_effect ε ε'pos : ε' > 0 f₀ : ℝ → ℂ ...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_of_neg_eq_conj_k	[10, 1]	[11, 76]	simp [k, Complex.conj_ofReal, ←Complex.exp_conj, Complex.conj_I, neg_mul]	x : ℝ ⊢ k (-x) = (starRingEnd ℂ) (k x)	no goals	Please generate a tactic in lean4 to solve the state. STATE: x : ℝ ⊢ k (-x) = (starRingEnd ℂ) (k x) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_of_abs_le_one	[13, 1]	[15, 12]	rw [k, max_eq_left (by linarith)]	x : ℝ abs_le_one : \|x\| ≤ 1 ⊢ k x = (1 - ↑\|x\|) / (1 - (Complex.I * ↑x).exp)	x : ℝ abs_le_one : \|x\| ≤ 1 ⊢ ↑(1 - \|x\|) / (1 - (Complex.I * ↑x).exp) = (1 - ↑\|x\|) / (1 - (Complex.I * ↑x).exp)	Please generate a tactic in lean4 to solve the state. STATE: x : ℝ abs_le_one : \|x\| ≤ 1 ⊢ k x = (1 - ↑\|x\|) / (1 - (Complex.I * ↑x).exp) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_of_abs_le_one	[13, 1]	[15, 12]	norm_cast	x : ℝ abs_le_one : \|x\| ≤ 1 ⊢ ↑(1 - \|x\|) / (1 - (Complex.I * ↑x).exp) = (1 - ↑\|x\|) / (1 - (Complex.I * ↑x).exp)	no goals	Please generate a tactic in lean4 to solve the state. STATE: x : ℝ abs_le_one : \|x\| ≤ 1 ⊢ ↑(1 - \|x\|) / (1 - (Complex.I * ↑x).exp) = (1 - ↑\|x\|) / (1 - (Complex.I * ↑x).exp) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_of_abs_le_one	[13, 1]	[15, 12]	linarith	x : ℝ abs_le_one : \|x\| ≤ 1 ⊢ 0 ≤ 1 - \|x\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: x : ℝ abs_le_one : \|x\| ≤ 1 ⊢ 0 ≤ 1 - \|x\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_of_one_le_abs	[17, 1]	[19, 7]	rw [k, max_eq_right (by linarith)]	x : ℝ abs_le_one : 1 ≤ \|x\| ⊢ k x = 0	x : ℝ abs_le_one : 1 ≤ \|x\| ⊢ ↑0 / (1 - (Complex.I * ↑x).exp) = 0	Please generate a tactic in lean4 to solve the state. STATE: x : ℝ abs_le_one : 1 ≤ \|x\| ⊢ k x = 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_of_one_le_abs	[17, 1]	[19, 7]	simp	x : ℝ abs_le_one : 1 ≤ \|x\| ⊢ ↑0 / (1 - (Complex.I * ↑x).exp) = 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: x : ℝ abs_le_one : 1 ≤ \|x\| ⊢ ↑0 / (1 - (Complex.I * ↑x).exp) = 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_of_one_le_abs	[17, 1]	[19, 7]	linarith	x : ℝ abs_le_one : 1 ≤ \|x\| ⊢ 1 - \|x\| ≤ 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: x : ℝ abs_le_one : 1 ≤ \|x\| ⊢ 1 - \|x\| ≤ 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_measurable	[21, 1]	[27, 18]	apply Measurable.div	⊢ Measurable k	case hf ⊢ Measurable fun a => ↑(max (1 - \|a\|) 0) case hg ⊢ Measurable fun a => 1 - (Complex.I * ↑a).exp	Please generate a tactic in lean4 to solve the state. STATE: ⊢ Measurable k TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_measurable	[21, 1]	[27, 18]	. measurability	case hg ⊢ Measurable fun a => 1 - (Complex.I * ↑a).exp	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hg ⊢ Measurable fun a => 1 - (Complex.I * ↑a).exp TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_measurable	[21, 1]	[27, 18]	apply Measurable.comp'	case hf ⊢ Measurable fun a => ↑(max (1 - \|a\|) 0)	case hf.hg ⊢ Measurable Complex.ofReal' case hf.hf ⊢ Measurable fun x => max (1 - \|x\|) 0 case hf.x ⊢ MeasurableSpace ℝ	Please generate a tactic in lean4 to solve the state. STATE: case hf ⊢ Measurable fun a => ↑(max (1 - \|a\|) 0) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_measurable	[21, 1]	[27, 18]	exact Complex.measurable_ofReal	case hf.hg ⊢ Measurable Complex.ofReal'	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hf.hg ⊢ Measurable Complex.ofReal' TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_measurable	[21, 1]	[27, 18]	exact Measurable.max (measurable_const.sub measurable_norm) measurable_const	case hf.hf ⊢ Measurable fun x => max (1 - \|x\|) 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hf.hf ⊢ Measurable fun x => max (1 - \|x\|) 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	k_measurable	[21, 1]	[27, 18]	measurability	case hg ⊢ Measurable fun a => 1 - (Complex.I * ↑a).exp	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hg ⊢ Measurable fun a => 1 - (Complex.I * ↑a).exp TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	by_cases h : 0 < \|x - y\| ∧ \|x - y\| < 1	x y : ℝ ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|)	case pos x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|) case neg x y : ℝ h : ¬(0 < \|x - y\| ∧ \|x - y\| < 1) ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|)	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	. calc ‖K x y‖ _ ≤ 1 / ‖1 - Complex.exp (Complex.I * ↑(x - y))‖ := by rw [K, k, norm_div] gcongr rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg] . apply max_le _ zero_le_one linarith [abs_nonneg (x-y)] . apply le_max_right _ ≤ 1 / (\|x - y\| / 2) := by gcongr ...	case pos x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|) case neg x y : ℝ h : ¬(0 < \|x - y\| ∧ \|x - y\| < 1) ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|)	case neg x y : ℝ h : ¬(0 < \|x - y\| ∧ \|x - y\| < 1) ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|)	Please generate a tactic in lean4 to solve the state. STATE: case pos x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|) case neg x y : ℝ h : ¬(0 < \|x - y\| ∧ \|x - y\| < 1) ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	. push_neg at h have : ‖K x y‖ = 0 := by rw [norm_eq_zero, K, k, _root_.div_eq_zero_iff] by_cases xeqy : x = y . right simp [xeqy] . left simp only [Complex.ofReal_eq_zero, max_eq_right_iff, tsub_le_iff_right, zero_add] exact h (abs_pos.mpr (sub_ne_zero.mpr xeqy)) rw [this] apply...	case neg x y : ℝ h : ¬(0 < \|x - y\| ∧ \|x - y\| < 1) ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case neg x y : ℝ h : ¬(0 < \|x - y\| ∧ \|x - y\| < 1) ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	calc ‖K x y‖ _ ≤ 1 / ‖1 - Complex.exp (Complex.I * ↑(x - y))‖ := by rw [K, k, norm_div] gcongr rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg] . apply max_le _ zero_le_one linarith [abs_nonneg (x-y)] . apply le_max_right _ ≤ 1 / (\|x - y\| / 2) := by gcongr . linarith . a...	case pos x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case pos x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	rw [K, k, norm_div]	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖K x y‖ ≤ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖↑(max (1 - \|x - y\|) 0)‖ / ‖1 - (Complex.I * ↑(x - y)).exp‖ ≤ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖K x y‖ ≤ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖ TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	gcongr	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖↑(max (1 - \|x - y\|) 0)‖ / ‖1 - (Complex.I * ↑(x - y)).exp‖ ≤ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖	case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖↑(max (1 - \|x - y\|) 0)‖ ≤ 1	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖↑(max (1 - \|x - y\|) 0)‖ / ‖1 - (Complex.I * ↑(x - y)).exp‖ ≤ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖ TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg]	case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖↑(max (1 - \|x - y\|) 0)‖ ≤ 1	case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ max (1 - \|x - y\|) 0 ≤ 1 case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 ≤ max (1 - \|x - y\|) 0	Please generate a tactic in lean4 to solve the state. STATE: case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ ‖↑(max (1 - \|x - y\|) 0)‖ ≤ 1 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	. apply max_le _ zero_le_one linarith [abs_nonneg (x-y)]	case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ max (1 - \|x - y\|) 0 ≤ 1 case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 ≤ max (1 - \|x - y\|) 0	case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 ≤ max (1 - \|x - y\|) 0	Please generate a tactic in lean4 to solve the state. STATE: case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ max (1 - \|x - y\|) 0 ≤ 1 case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 ≤ max (1 - \|x - y\|) 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	. apply le_max_right	case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 ≤ max (1 - \|x - y\|) 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 ≤ max (1 - \|x - y\|) 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	apply max_le _ zero_le_one	case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ max (1 - \|x - y\|) 0 ≤ 1	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 1 - \|x - y\| ≤ 1	Please generate a tactic in lean4 to solve the state. STATE: case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ max (1 - \|x - y\|) 0 ≤ 1 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	linarith [abs_nonneg (x-y)]	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 1 - \|x - y\| ≤ 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 1 - \|x - y\| ≤ 1 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	apply le_max_right	case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 ≤ max (1 - \|x - y\|) 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hab x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 ≤ max (1 - \|x - y\|) 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	gcongr	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖ ≤ 1 / (\|x - y\| / 2)	case hc x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 < \|x - y\| / 2 case h x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖ ≤ 1 / (\|x - y\| / 2) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	. linarith	case hc x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 < \|x - y\| / 2 case h x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖	case h x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖	Please generate a tactic in lean4 to solve the state. STATE: case hc x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 < \|x - y\| / 2 case h x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖ TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	. apply lower_secant_bound _ (by rfl) rw [Set.mem_Icc] constructor . simp calc \|x - y\| _ ≤ 1 := h.2.le _ ≤ 2 * Real.pi - 1 := by rw [le_sub_iff_add_le] linarith [Real.two_le_pi] _ ≤ 2 * Real.pi + (x - y) := by rw [sub_eq_add_neg] gcongr exact (abs_le.m...	case h x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖ TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	linarith	case hc x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 < \|x - y\| / 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hc x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 0 < \|x - y\| / 2 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	apply lower_secant_bound _ (by rfl)	case h x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ x - y ∈ Set.Icc (-2 * Real.pi + \|x - y\|) (2 * Real.pi - \|x - y\|)	Please generate a tactic in lean4 to solve the state. STATE: case h x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖ TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	rw [Set.mem_Icc]	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ x - y ∈ Set.Icc (-2 * Real.pi + \|x - y\|) (2 * Real.pi - \|x - y\|)	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ -2 * Real.pi + \|x - y\| ≤ x - y ∧ x - y ≤ 2 * Real.pi - \|x - y\|	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ x - y ∈ Set.Icc (-2 * Real.pi + \|x - y\|) (2 * Real.pi - \|x - y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	constructor	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ -2 * Real.pi + \|x - y\| ≤ x - y ∧ x - y ≤ 2 * Real.pi - \|x - y\|	case left x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ -2 * Real.pi + \|x - y\| ≤ x - y case right x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ x - y ≤ 2 * Real.pi - \|x - y\|	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ -2 * Real.pi + \|x - y\| ≤ x - y ∧ x - y ≤ 2 * Real.pi - \|x - y\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	. simp calc \|x - y\| _ ≤ 1 := h.2.le _ ≤ 2 * Real.pi - 1 := by rw [le_sub_iff_add_le] linarith [Real.two_le_pi] _ ≤ 2 * Real.pi + (x - y) := by rw [sub_eq_add_neg] gcongr exact (abs_le.mp h.2.le).1	case left x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ -2 * Real.pi + \|x - y\| ≤ x - y case right x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ x - y ≤ 2 * Real.pi - \|x - y\|	case right x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ x - y ≤ 2 * Real.pi - \|x - y\|	Please generate a tactic in lean4 to solve the state. STATE: case left x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ -2 * Real.pi + \|x - y\| ≤ x - y case right x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ x - y ≤ 2 * Real.pi - \|x - y\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	. calc x - y _ ≤ \|x - y\| := le_abs_self (x - y) _ ≤ 1 := h.2.le _ ≤ 2 * Real.pi - 1 := by rw [le_sub_iff_add_le] linarith [Real.two_le_pi] _ ≤ 2 * Real.pi - \|x - y\| := by gcongr exact h.2.le	case right x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ x - y ≤ 2 * Real.pi - \|x - y\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ x - y ≤ 2 * Real.pi - \|x - y\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	rfl	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| ≤ \|x - y\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| ≤ \|x - y\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	simp	case left x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ -2 * Real.pi + \|x - y\| ≤ x - y	case left x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| ≤ 2 * Real.pi + (x - y)	Please generate a tactic in lean4 to solve the state. STATE: case left x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ -2 * Real.pi + \|x - y\| ≤ x - y TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	calc \|x - y\| _ ≤ 1 := h.2.le _ ≤ 2 * Real.pi - 1 := by rw [le_sub_iff_add_le] linarith [Real.two_le_pi] _ ≤ 2 * Real.pi + (x - y) := by rw [sub_eq_add_neg] gcongr exact (abs_le.mp h.2.le).1	case left x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| ≤ 2 * Real.pi + (x - y)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case left x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| ≤ 2 * Real.pi + (x - y) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	rw [le_sub_iff_add_le]	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 1 ≤ 2 * Real.pi - 1	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 1 + 1 ≤ 2 * Real.pi	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 1 ≤ 2 * Real.pi - 1 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	linarith [Real.two_le_pi]	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 1 + 1 ≤ 2 * Real.pi	no goals	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 1 + 1 ≤ 2 * Real.pi TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	rw [sub_eq_add_neg]	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 2 * Real.pi - 1 ≤ 2 * Real.pi + (x - y)	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 2 * Real.pi + -1 ≤ 2 * Real.pi + (x - y)	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 2 * Real.pi - 1 ≤ 2 * Real.pi + (x - y) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	gcongr	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 2 * Real.pi + -1 ≤ 2 * Real.pi + (x - y)	case bc x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ -1 ≤ x - y	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 2 * Real.pi + -1 ≤ 2 * Real.pi + (x - y) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	exact (abs_le.mp h.2.le).1	case bc x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ -1 ≤ x - y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case bc x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ -1 ≤ x - y TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	calc x - y _ ≤ \|x - y\| := le_abs_self (x - y) _ ≤ 1 := h.2.le _ ≤ 2 * Real.pi - 1 := by rw [le_sub_iff_add_le] linarith [Real.two_le_pi] _ ≤ 2 * Real.pi - \|x - y\| := by gcongr exact h.2.le	case right x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ x - y ≤ 2 * Real.pi - \|x - y\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ x - y ≤ 2 * Real.pi - \|x - y\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	gcongr	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 2 * Real.pi - 1 ≤ 2 * Real.pi - \|x - y\|	case h x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| ≤ 1	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 2 * Real.pi - 1 ≤ 2 * Real.pi - \|x - y\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	exact h.2.le	case h x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| ≤ 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\| ≤ 1 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	field_simp	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 1 / (\|x - y\| / 2) = 2 / \|x - y\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 1 / (\|x - y\| / 2) = 2 / \|x - y\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	ring_nf	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 2 / \|x - y\| ≤ 2 ^ 2 / (2 * \|x - y\|)	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\|⁻¹ * 2 ≤ \|x - y\|⁻¹ * 2	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ 2 / \|x - y\| ≤ 2 ^ 2 / (2 * \|x - y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	trivial	x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\|⁻¹ * 2 ≤ \|x - y\|⁻¹ * 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| ∧ \|x - y\| < 1 ⊢ \|x - y\|⁻¹ * 2 ≤ \|x - y\|⁻¹ * 2 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	push_neg at h	case neg x y : ℝ h : ¬(0 < \|x - y\| ∧ \|x - y\| < 1) ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|)	case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|)	Please generate a tactic in lean4 to solve the state. STATE: case neg x y : ℝ h : ¬(0 < \|x - y\| ∧ \|x - y\| < 1) ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	have : ‖K x y‖ = 0 := by rw [norm_eq_zero, K, k, _root_.div_eq_zero_iff] by_cases xeqy : x = y . right simp [xeqy] . left simp only [Complex.ofReal_eq_zero, max_eq_right_iff, tsub_le_iff_right, zero_add] exact h (abs_pos.mpr (sub_ne_zero.mpr xeqy))	case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|)	case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|)	Please generate a tactic in lean4 to solve the state. STATE: case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	rw [this]	case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|)	case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 ^ 2 / (2 * \|x - y\|)	Please generate a tactic in lean4 to solve the state. STATE: case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * \|x - y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	apply div_nonneg	case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 ^ 2 / (2 * \|x - y\|)	case neg.ha x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 ^ 2 case neg.hb x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 * \|x - y\|	Please generate a tactic in lean4 to solve the state. STATE: case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 ^ 2 / (2 * \|x - y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	. norm_num	case neg.ha x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 ^ 2 case neg.hb x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 * \|x - y\|	case neg.hb x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 * \|x - y\|	Please generate a tactic in lean4 to solve the state. STATE: case neg.ha x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 ^ 2 case neg.hb x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 * \|x - y\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	. linarith [abs_nonneg (x-y)]	case neg.hb x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 * \|x - y\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: case neg.hb x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 * \|x - y\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	rw [norm_eq_zero, K, k, _root_.div_eq_zero_iff]	x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| ⊢ ‖K x y‖ = 0	x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| ⊢ ‖K x y‖ = 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	by_cases xeqy : x = y	x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0	case pos x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0	Please generate a tactic in lean4 to solve the state. STATE: x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	. right simp [xeqy]	case pos x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0	case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0	Please generate a tactic in lean4 to solve the state. STATE: case pos x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 TACTI...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	. left simp only [Complex.ofReal_eq_zero, max_eq_right_iff, tsub_le_iff_right, zero_add] exact h (abs_pos.mpr (sub_ne_zero.mpr xeqy))	case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	right	case pos x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0	case pos.h x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : x = y ⊢ 1 - (Complex.I * ↑(x - y)).exp = 0	Please generate a tactic in lean4 to solve the state. STATE: case pos x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	simp [xeqy]	case pos.h x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : x = y ⊢ 1 - (Complex.I * ↑(x - y)).exp = 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case pos.h x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : x = y ⊢ 1 - (Complex.I * ↑(x - y)).exp = 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	left	case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0	case neg.h x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0	Please generate a tactic in lean4 to solve the state. STATE: case neg x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	simp only [Complex.ofReal_eq_zero, max_eq_right_iff, tsub_le_iff_right, zero_add]	case neg.h x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0	case neg.h x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ 1 ≤ \|x - y\|	Please generate a tactic in lean4 to solve the state. STATE: case neg.h x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ ↑(max (1 - \|x - y\|) 0) = 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	exact h (abs_pos.mpr (sub_ne_zero.mpr xeqy))	case neg.h x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ 1 ≤ \|x - y\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: case neg.h x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| xeqy : ¬x = y ⊢ 1 ≤ \|x - y\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	norm_num	case neg.ha x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 ^ 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: case neg.ha x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 ^ 2 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_bound	[37, 1]	[90, 34]	linarith [abs_nonneg (x-y)]	case neg.hb x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 * \|x - y\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: case neg.hb x y : ℝ h : 0 < \|x - y\| → 1 ≤ \|x - y\| this : ‖K x y‖ = 0 ⊢ 0 ≤ 2 * \|x - y\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Real.volume_uIoc	[93, 1]	[95, 48]	rw [Set.uIoc, volume_Ioc, max_sub_min_eq_abs]	a b : ℝ ⊢ MeasureTheory.volume (Ι a b) = ENNReal.ofReal \|b - a\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: a b : ℝ ⊢ MeasureTheory.volume (Ι a b) = ENNReal.ofReal \|b - a\| TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	rw [k_of_abs_le_one, k_of_abs_le_one]	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 6 * (1 / \|y\|) * (\|y - y'\| / \|y\|)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖(1 - ↑\|(-y)\|) / (1 - (Complex.I * ↑(-y)).exp) - (1 - ↑\|(-y')\|) / (1 - (Complex.I * ↑(-y')).exp)‖ ≤ 2 ^ 6 * (1 / \|y\|) * (\|y - y'\| / \|y\|) y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy'...	Please generate a tactic in lean4 to solve the state. STATE: y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 6 * (1 / \|y\|) * (\|y - y'\| / \|y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	. simp only [abs_neg, Complex.ofReal_neg, mul_neg, ge_iff_le] rw [abs_of_nonneg yy'nonneg.1, abs_of_nonneg yy'nonneg.2] let f : ℝ → ℂ := fun t ↦ (1 - t) / (1 - Complex.exp (-(Complex.I * t))) set f' : ℝ → ℂ := fun t ↦ (-1 + Complex.exp (-(Complex.I * t)) + Complex.I * (t - 1) * Complex.exp (-(Complex.I * t))) / (...	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖(1 - ↑\|(-y)\|) / (1 - (Complex.I * ↑(-y)).exp) - (1 - ↑\|(-y')\|) / (1 - (Complex.I * ↑(-y')).exp)‖ ≤ 2 ^ 6 * (1 / \|y\|) * (\|y - y'\| / \|y\|) y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy'...	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ \|(-y')\| ≤ 1 y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ \|(-y)\| ≤ 1	Please generate a tactic in lean4 to solve the state. STATE: y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖(1 - ↑\|(-y)\|) / (1 - (Complex.I * ↑(-y)).exp) - (1 - ↑\|(-y')\|) / (1 - (Complex.I * ↑(-y')).exp)‖ ≤ 2 ^ 6 * (1 / \|y\|) * (\|y - y'\| / \|y\|) y y' : ℝ yy'nonneg : 0...
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	. rw [abs_neg, abs_of_nonneg yy'nonneg.2] assumption	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ \|(-y')\| ≤ 1 y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ \|(-y)\| ≤ 1	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ \|(-y)\| ≤ 1	Please generate a tactic in lean4 to solve the state. STATE: y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ \|(-y')\| ≤ 1 y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ \|(-y)\| ≤ 1 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	. rw [abs_neg, abs_of_nonneg yy'nonneg.1] assumption	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ \|(-y)\| ≤ 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ \|(-y)\| ≤ 1 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	simp only [abs_neg, Complex.ofReal_neg, mul_neg, ge_iff_le]	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖(1 - ↑\|(-y)\|) / (1 - (Complex.I * ↑(-y)).exp) - (1 - ↑\|(-y')\|) / (1 - (Complex.I * ↑(-y')).exp)‖ ≤ 2 ^ 6 * (1 / \|y\|) * (\|y - y'\| / \|y\|)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖(1 - ↑\|y\|) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑\|y'\|) / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / \|y\|) * (\|y - y'\| / \|y\|)	Please generate a tactic in lean4 to solve the state. STATE: y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖(1 - ↑\|(-y)\|) / (1 - (Complex.I * ↑(-y)).exp) - (1 - ↑\|(-y')\|) / (1 - (Complex.I * ↑(-y')).exp)‖ ≤ 2 ^ 6 * (1 / \|y\|) * (\|y - y'\| / \|y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	rw [abs_of_nonneg yy'nonneg.1, abs_of_nonneg yy'nonneg.2]	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖(1 - ↑\|y\|) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑\|y'\|) / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / \|y\|) * (\|y - y'\| / \|y\|)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y)	Please generate a tactic in lean4 to solve the state. STATE: y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖(1 - ↑\|y\|) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑\|y'\|) / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / \|y\|) * (\|y - y'\| / \|y\|) TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	let f : ℝ → ℂ := fun t ↦ (1 - t) / (1 - Complex.exp (-(Complex.I * t)))	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y)	Please generate a tactic in lean4 to solve the state. STATE: y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y) TACTIC:

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