| import OAI.Probability.ClassicalON.DyadicSine |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators ComplexConjugate |
| namespace ClassicalON |
|
|
| variable {k : ℕ} |
|
|
| def groupedPoint (m s : Bool) (p : DyadicFreq k) : ℤ × ℤ := |
| let q := if m then (dyadicPoint p).swap else dyadicPoint p |
| if s then -q else q |
|
|
| def groupedTorusPoint (m s : Bool) (p : DyadicFreq k) : DiscreteTorus (16*2^k) := |
| (((groupedPoint m s p).1:ZMod (16*2^k)), ((groupedPoint m s p).2:ZMod (16*2^k))) |
| |
| instance dyadicModulus_neZero (k : ℕ) : NeZero (16*2^k) := ⟨by positivity⟩ |
| |
| theorem groupedTorusPoint_injective (m s : Bool) : |
| Function.Injective (groupedTorusPoint (k := k) m s) := by |
| have hs := dyadicTorusPoint_injective (k := k) |
| cases m <;> cases s <;> intro p q h <;> apply hs |
| · exact h |
| · simpa [groupedTorusPoint, groupedPoint, dyadicTorusPoint, Prod.ext_iff] using h |
| · exact Prod.ext (congrArg Prod.snd h) (congrArg Prod.fst h) |
| · have h' : (dyadicTorusPoint p).swap = (dyadicTorusPoint q).swap := by |
| simpa [groupedTorusPoint, groupedPoint, dyadicTorusPoint, Prod.ext_iff] using h |
| exact Prod.swap_injective h' |
| |
| def dyadicWeight (p : DyadicFreq k) : ℝ := |
| 1/((dyadicRadius p:ℝ)*Real.sqrt (dyadicRadius p:ℝ)) |
| |
| theorem dyadicWeight_pos (p : DyadicFreq k) : 0 < dyadicWeight p := by |
| have h : (0:ℝ)<dyadicRadius p := by exact_mod_cast dyadicRadius_pos p |
| unfold dyadicWeight |
| positivity |
| |
| @[simp] theorem dyadicWeight_sq (p : DyadicFreq k) : |
| dyadicWeight p^2 = (1/(dyadicRadius p:ℝ))^3 := by |
| have hr : (0:ℝ)<dyadicRadius p := by exact_mod_cast dyadicRadius_pos p |
| unfold dyadicWeight |
| rw [div_pow, mul_pow, Real.sq_sqrt hr.le] |
| field_simp |
| |
| @[simp] theorem dyadicWeight_mul_sqrt (p : DyadicFreq k) : |
| dyadicWeight p*Real.sqrt (dyadicRadius p:ℝ) = 1/(dyadicRadius p:ℝ) := by |
| have hr : (0:ℝ)<dyadicRadius p := by exact_mod_cast dyadicRadius_pos p |
| have hs : Real.sqrt (dyadicRadius p:ℝ) ≠ 0 := (Real.sqrt_pos.mpr hr).ne' |
| unfold dyadicWeight |
| field_simp |
| |
| theorem wave_commutator (a b c θ : ℝ) (w : ℂ) (hw : ‖w‖=1) : |
| ((a*b:ℝ):ℂ)*w * conj (((a*c:ℝ):ℂ)*w*Complex.exp (Complex.I*(θ:ℂ))) / 2 - |
| conj (((a*b:ℝ):ℂ)*w) * (((a*c:ℝ):ℂ)*w*Complex.exp (Complex.I*(θ:ℂ))) / 2 = |
| -Complex.I * ((a^2*b*c*Real.sin θ:ℝ):ℂ) := by |
| have hww : w*conj w=1 := by rw [Complex.mul_conj, Complex.normSq_eq_norm_sq, hw]; norm_num |
| rw [mul_comm Complex.I (θ:ℂ), Complex.exp_ofReal_mul_I] |
| simp only [map_mul, map_add, Complex.conj_ofReal, Complex.conj_I] |
| push_cast |
| linear_combination -Complex.I*(a:ℂ)^2*(b:ℂ)*(c:ℂ)*(Complex.sin (θ:ℂ)) * hww |
| |
| end ClassicalON |
| |
| end |
| |
| end OAI |
| |
