File size: 3,116 Bytes
fac6777 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileCertificateComparisons
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.ParameterOpenness
namespace OAI
/-! The real-linear map certified at the free profile center. -/
namespace DefocusingNLS.ProfileCertificate
noncomputable def realLinearMap (x y : ℂ) : ℂ →L[ℝ] ℂ :=
Complex.reCLM.smulRight x + Complex.imCLM.smulRight y
@[simp] theorem realLinearMap_apply (x y z : ℂ) :
realLinearMap x y z = (z.re : ℂ)*x+(z.im : ℂ)*y := by
simp [realLinearMap, Complex.real_smul]
/-- The two strict rational Gram inequalities give the Euclidean lower bound. -/
theorem realLinearMap_minimum_stretch (x y : ℂ) (c : ℝ) (hc : 0 ≤ c)
(hx : c^2 < Complex.normSq x)
(hgram : (Complex.normSq x-c^2)*(Complex.normSq y-c^2) >
(x.re*y.re+x.im*y.im)^2) (z : ℂ) :
c*‖z‖ ≤ ‖realLinearMap x y z‖ := by
let A := Complex.normSq x-c^2
let B := Complex.normSq y-c^2
let C := x.re*y.re+x.im*y.im
have hA : 0 < A := sub_pos.mpr hx
have hD : 0 ≤ A*B-C^2 := le_of_lt (sub_pos.mpr hgram)
have he : A*(Complex.normSq (realLinearMap x y z)-c^2*Complex.normSq z) =
(A*z.re+C*z.im)^2+(A*B-C^2)*z.im^2 := by
simp only [realLinearMap_apply, Complex.normSq_apply, Complex.add_re,
Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.ofReal_re,
Complex.ofReal_im, zero_mul, sub_zero, A, B, C]
ring
have hnon : 0 ≤ A*(Complex.normSq (realLinearMap x y z)-c^2*Complex.normSq z) := by
rw [he]
exact add_nonneg (sq_nonneg _) (mul_nonneg hD (sq_nonneg _))
have hq := nonneg_of_mul_nonneg_right hnon hA
rw [Complex.normSq_eq_norm_sq, Complex.normSq_eq_norm_sq] at hq
have hcz : 0 ≤ c*‖z‖ := mul_nonneg hc (norm_nonneg _)
nlinarith [norm_nonneg (realLinearMap x y z)]
noncomputable def centralX : ℂ :=
RationalComplex.toComplex (RationalComplex.quotient result.derivB.d result.value.b)
noncomputable def centralY : ℂ :=
RationalComplex.toComplex (RationalComplex.quotient result.derivZ.d result.value.b)
/-- The concrete map has the precise lower bound needed in the degree argument. -/
theorem central_minimum_stretch (z : ℂ) :
(48/1000 : ℝ)*‖z‖ ≤ ‖realLinearMap centralX centralY z‖ := by
have h := linear_bounds
rcases h with ⟨hx, hgram, _⟩
apply realLinearMap_minimum_stretch centralX centralY (48/1000) (by norm_num) _ _ z
· simp only [centralX, RationalComplex.normSq_toComplex]
have hh := (Rat.cast_lt (K := ℝ)).mpr hx
push_cast at hh ⊢
exact hh
· simp only [centralX, centralY, RationalComplex.normSq_toComplex,
RationalComplex.toComplex_re, RationalComplex.toComplex_im]
have hh := (Rat.cast_lt (K := ℝ)).mpr hgram
push_cast at hh ⊢
exact hh
theorem central_linear_orientation :
centralX.re*centralY.im-centralX.im*centralY.re < 0 := by
have h := linear_bounds
rcases h with ⟨_, _, _, _, _, _, hdet⟩
simp only [centralX, centralY, RationalComplex.toComplex_re, RationalComplex.toComplex_im]
exact_mod_cast hdet
end DefocusingNLS.ProfileCertificate
end OAI
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