| import Mathlib |
| import OAI.Analysis.Conductivity.Model |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology MeasureTheory Matrix |
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| theorem leading_frequency_SL2_alignment {h : Fin 2 → ℤ} (hh : h≠0) : |
| ∃ (k : ℤ) (A : Matrix (Fin 2) (Fin 2) ℤ), |
| 0<k ∧ A.det=1 ∧ h ᵥ* A=![0,k] := by |
| have hg : 0<Int.gcd (h 0) (h 1) := by |
| apply Nat.pos_of_ne_zero |
| intro he |
| have hz := Int.gcd_eq_zero_iff.mp he |
| apply hh |
| ext i |
| fin_cases i |
| · exact hz.1 |
| · exact hz.2 |
| obtain ⟨u,v,huv,h0,h1⟩ := Int.exists_gcd_one hg |
| let a := Int.gcdA u v |
| let b := Int.gcdB u v |
| have hab : u*a+v*b=1 := by |
| simpa only [huv,Int.natCast_one] using (Int.gcd_eq_gcd_ab u v).symm |
| refine ⟨Int.gcd (h 0) (h 1),!![v,a;-u,b],by exact_mod_cast hg,?_,?_⟩ |
| · rw [Matrix.det_fin_two_of] |
| nlinarith |
| · ext i |
| fin_cases i |
| · simp only [Matrix.vecMul,dotProduct,Fin.sum_univ_two,Matrix.of_apply, |
| Matrix.cons_val_zero,Matrix.cons_val_one] |
| change h 0*v+h 1*(-u)=0 |
| calc |
| _ = (u*(Int.gcd (h 0) (h 1):ℤ))*v+(v*(Int.gcd (h 0) (h 1):ℤ))*(-u) := |
| congrArg₂ (fun x y : ℤ => x*v+y*(-u)) h0 h1 |
| _ = 0 := by ring |
| · simp only [Matrix.vecMul,dotProduct,Fin.sum_univ_two,Matrix.of_apply, |
| Matrix.cons_val_zero,Matrix.cons_val_one] |
| change h 0*a+h 1*b=(Int.gcd (h 0) (h 1):ℤ) |
| calc |
| _ = (u*a+v*b)*(Int.gcd (h 0) (h 1):ℤ) := by |
| calc |
| _ = (u*(Int.gcd (h 0) (h 1):ℤ))*a+(v*(Int.gcd (h 0) (h 1):ℤ))*b := |
| congrArg₂ (fun x y : ℤ => x*a+y*b) h0 h1 |
| _ = _ := by ring |
| _ = _ := by rw [hab,one_mul] |
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| end ScalarConductivity |
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| namespace ScalarConductivity |
| open Set Filter Topology MeasureTheory Matrix |
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| def angularIntMap (A : Matrix (Fin 2) (Fin 2) ℤ) : |
| UnitAddTorus (Fin 2) →+ UnitAddTorus (Fin 2) where |
| toFun x := ![A 0 0 • x 0+A 0 1 • x 1,A 1 0 • x 0+A 1 1 • x 1] |
| map_zero' := by ext i; fin_cases i <;> simp |
| map_add' x y := by ext i; fin_cases i <;> simp [smul_add] <;> abel |
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| lemma angularIntMap_continuous (A : Matrix (Fin 2) (Fin 2) ℤ) : |
| Continuous (angularIntMap A) := by |
| apply continuous_pi |
| intro i |
| fin_cases i <;> |
| exact (continuous_apply 0 |>.zsmul _).add (continuous_apply 1 |>.zsmul _) |
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| lemma angularIntMap_surjective {A : Matrix (Fin 2) (Fin 2) ℤ} (hA : A.det=1) : |
| Function.Surjective (angularIntMap A) := by |
| have hd : A 0 0*A 1 1-A 0 1*A 1 0=1 := by simpa only [Matrix.det_fin_two] using hA |
| intro y |
| refine ⟨![A 1 1 • y 0- A 0 1 • y 1,-A 1 0 • y 0+A 0 0 • y 1],?_⟩ |
| ext i |
| fin_cases i |
| · change A 0 0 • (A 1 1 • y 0-A 0 1 • y 1)+ |
| A 0 1 • (-A 1 0 • y 0+A 0 0 • y 1)=y 0 |
| calc |
| _ = (A 0 0*A 1 1-A 0 1*A 1 0) • y 0 := by module |
| _ = _ := by rw [hd,one_smul] |
| · change A 1 0 • (A 1 1 • y 0-A 0 1 • y 1)+ |
| A 1 1 • (-A 1 0 • y 0+A 0 0 • y 1)=y 1 |
| calc |
| _ = (A 0 0*A 1 1-A 0 1*A 1 0) • y 1 := by module |
| _ = _ := by rw [hd,one_smul] |
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| lemma angularIntMap_measurePreserving {A : Matrix (Fin 2) (Fin 2) ℤ} (hA : A.det=1) : |
| MeasurePreserving (angularIntMap A) (volume : Measure (UnitAddTorus (Fin 2))) volume := |
| AddMonoidHom.measurePreserving (angularIntMap_continuous A) (angularIntMap_surjective hA) rfl |
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| end ScalarConductivity |
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| namespace ScalarConductivity |
| open Set Filter Topology MeasureTheory Matrix |
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| def angularNormalization (A : Matrix (Fin 2) (Fin 2) ℤ) : ℝ := |
| (A 0 0:ℝ)^2+(A 0 1:ℝ)^2+(A 1 0:ℝ)^2+(A 1 1:ℝ)^2+1 |
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| def normalizedAngularTensor (s : Fin 3 → ℝ) (A : Matrix (Fin 2) (Fin 2) ℤ) : Fin 3 → ℝ := |
| let a : ℝ := A 0 0 |
| let b : ℝ := A 0 1 |
| let c : ℝ := A 1 0 |
| let d : ℝ := A 1 1 |
| let M := angularNormalization A |
| ![M*(s 0*d^2-2*s 1*b*d+s 2*b^2), |
| M*(-s 0*c*d+s 1*(a*d+b*c)-s 2*a*b), |
| M*(s 0*c^2-2*s 1*a*c+s 2*a^2)] |
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| lemma angularNormalization_pos (A : Matrix (Fin 2) (Fin 2) ℤ) : |
| 0<angularNormalization A := by unfold angularNormalization; positivity |
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| lemma twoCoordinate_cauchy (a b x y : ℝ) : |
| (a*x+b*y)^2≤(a^2+b^2)*(x^2+y^2) := by |
| nlinarith [sq_nonneg (a*y-b*x)] |
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| theorem normalizedAngularTensor_lower {s : Fin 3 → ℝ} |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2)≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| {A : Matrix (Fin 2) (Fin 2) ℤ} (hA : A.det=1) (x y : ℝ) : |
| (1/2)*(x^2+y^2)≤normalizedAngularTensor s A 0*x^2+ |
| 2*normalizedAngularTensor s A 1*x*y+normalizedAngularTensor s A 2*y^2 := by |
| let a : ℝ := A 0 0 |
| let b : ℝ := A 0 1 |
| let c : ℝ := A 1 0 |
| let d : ℝ := A 1 1 |
| let M := angularNormalization A |
| have hdZ : A 0 0*A 1 1-A 0 1*A 1 0=1 := by |
| simpa only [Matrix.det_fin_two] using hA |
| have hd : a*d-b*c=1 := by |
| dsimp [a,b,c,d] |
| exact_mod_cast hdZ |
| let u := d*x-c*y |
| let v := -b*x+a*y |
| have hx : a*u+c*v=x := by |
| calc |
| _ = (a*d-b*c)*x := by dsimp [u,v]; ring |
| _ = _ := by rw [hd,one_mul] |
| have hy : b*u+d*v=y := by |
| calc |
| _ = (a*d-b*c)*y := by dsimp [u,v]; ring |
| _ = _ := by rw [hd,one_mul] |
| have hn : x^2+y^2≤M*(u^2+v^2) := by |
| calc |
| _ = (a*u+c*v)^2+(b*u+d*v)^2 := by rw [hx,hy] |
| _ ≤ (a^2+c^2)*(u^2+v^2)+(b^2+d^2)*(u^2+v^2) := |
| add_le_add (twoCoordinate_cauchy a c u v) (twoCoordinate_cauchy b d u v) |
| _ ≤ M*(u^2+v^2) := by |
| dsimp [M,angularNormalization,a,b,c,d] |
| nlinarith [sq_nonneg u,sq_nonneg v] |
| have he : normalizedAngularTensor s A 0*x^2+ |
| 2*normalizedAngularTensor s A 1*x*y+normalizedAngularTensor s A 2*y^2= |
| M*(s 0*u^2+2*s 1*u*v+s 2*v^2) := by |
| dsimp [normalizedAngularTensor,M,u,v,a,b,c,d] |
| ring |
| rw [he] |
| have hm := mul_le_mul_of_nonneg_left (hs u v) (angularNormalization_pos A).le |
| nlinarith |
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| end ScalarConductivity |
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| end |
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| end OAI |
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