openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticContinuationRegular.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticContinuationSpatial | |
| namespace OAI | |
| /-! Measurability and finite spatial derivative bounds for the actual | |
| Gaussian continuation formulas through order three. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron | |
| open scoped NNReal | |
| namespace InvariantIsing | |
| def MagneticContinuationBound (f : ℝ → ℝ) : Prop := | |
| ∃ K : ℝ, 0 ≤ K ∧ ∀ x, |f x| ≤ K | |
| namespace MagneticContinuationBound | |
| lemma const (c : ℝ) : MagneticContinuationBound (fun _ => c) := | |
| ⟨|c|, abs_nonneg _, fun _ => le_rfl⟩ | |
| lemma add {f g : ℝ → ℝ} (hf : MagneticContinuationBound f) | |
| (hg : MagneticContinuationBound g) : MagneticContinuationBound (fun x => f x + g x) := by | |
| obtain ⟨K, hK, bK⟩ := hf | |
| obtain ⟨L, hL, bL⟩ := hg | |
| exact ⟨K + L, add_nonneg hK hL, fun x => | |
| (abs_add_le _ _).trans (add_le_add (bK x) (bL x))⟩ | |
| lemma sub {f g : ℝ → ℝ} (hf : MagneticContinuationBound f) | |
| (hg : MagneticContinuationBound g) : MagneticContinuationBound (fun x => f x - g x) := by | |
| obtain ⟨K, hK, bK⟩ := hf | |
| obtain ⟨L, hL, bL⟩ := hg | |
| exact ⟨K + L, add_nonneg hK hL, fun x => | |
| (abs_sub _ _).trans (add_le_add (bK x) (bL x))⟩ | |
| lemma mul {f g : ℝ → ℝ} (hf : MagneticContinuationBound f) | |
| (hg : MagneticContinuationBound g) : MagneticContinuationBound (fun x => f x * g x) := by | |
| obtain ⟨K, hK, bK⟩ := hf | |
| obtain ⟨L, hL, bL⟩ := hg | |
| refine ⟨K * L, mul_nonneg hK hL, fun x => ?_⟩ | |
| rw [abs_mul] | |
| exact mul_le_mul (bK x) (bL x) (abs_nonneg _) hK | |
| lemma transition (ζ : ℝ) (v : ℝ≥0) {F A : ℝ → ℝ} | |
| (hF : Measurable F) (hG : HasLinearGrowth F) (hA : MagneticContinuationBound A) : | |
| MagneticContinuationBound (fieldSpinTransition ζ v F A) := by | |
| obtain ⟨K, hK, bK⟩ := hA | |
| exact ⟨K, hK, fieldSpinTransition_bound ζ v hF hG bK⟩ | |
| lemma spatial (ζ : ℝ) (v : ℝ≥0) {F M A B : ℝ → ℝ} | |
| (hF : Measurable F) (hG : HasLinearGrowth F) | |
| (hM : MagneticContinuationBound M) (hA : MagneticContinuationBound A) | |
| (hB : MagneticContinuationBound B) : | |
| MagneticContinuationBound (fieldTiltSpatial ζ v F M A B) := by | |
| obtain ⟨K, hK, bK⟩ := hM | |
| obtain ⟨L, hL, bL⟩ := hA | |
| obtain ⟨C, hC, bC⟩ := hB | |
| exact ⟨C + 2 * |ζ| * L * K, by positivity, | |
| fieldSpinTransition_derivative_bound ζ v hF hG hK hL bK bL bC⟩ | |
| end MagneticContinuationBound | |
| lemma measurable_magneticContinuationSecond (ζ : ℝ) (v : ℝ≥0) | |
| {F M Q A B C : ℝ → ℝ} (hF : Measurable F) | |
| (hM : Measurable M) (hQ : Measurable Q) | |
| (hA : Measurable A) (hB : Measurable B) (hC : Measurable C) : | |
| Measurable (magneticContinuationSecond ζ v F M Q A B C) := by | |
| have hS := measurable_fieldTiltSpatial ζ v hF hM hB hC | |
| have hSA := measurable_fieldTiltSpatial ζ v hF hM hA hB | |
| have hSM := measurable_fieldTiltSpatial ζ v hF hM hM hQ | |
| have hSAM := measurable_fieldTiltSpatial ζ v hF hM (hA.mul hM) | |
| ((hB.mul hM).add (hA.mul hQ)) | |
| have hTA := measurable_fieldSpinTransition ζ v hF hA | |
| have hTM := measurable_fieldSpinTransition ζ v hF hM | |
| exact hS.add ((hSAM.sub ((hSA.mul hTM).add (hTA.mul hSM))).const_mul ζ) | |
| lemma magneticContinuationSecond_bounded (ζ : ℝ) (v : ℝ≥0) | |
| {F M Q A B C : ℝ → ℝ} (hF : Measurable F) (hG : HasLinearGrowth F) | |
| (hM : MagneticContinuationBound M) (hQ : MagneticContinuationBound Q) | |
| (hA : MagneticContinuationBound A) (hB : MagneticContinuationBound B) | |
| (hC : MagneticContinuationBound C) : | |
| MagneticContinuationBound (magneticContinuationSecond ζ v F M Q A B C) := by | |
| have T {A : ℝ → ℝ} (hA : MagneticContinuationBound A) := | |
| MagneticContinuationBound.transition ζ v hF hG hA | |
| have D {A B : ℝ → ℝ} (hA : MagneticContinuationBound A) (hB : MagneticContinuationBound B) := | |
| MagneticContinuationBound.spatial ζ v hF hG hM hA hB | |
| exact (D hB hC).add ((MagneticContinuationBound.const ζ).mul | |
| ((D (hA.mul hM) ((hB.mul hM).add (hA.mul hQ))).sub | |
| (((D hA hB).mul (T hM)).add ((T hA).mul (D hM hQ))))) | |
| lemma measurable_magneticContinuationThird (ζ : ℝ) (v : ℝ≥0) | |
| {F M Q R A B C D : ℝ → ℝ} (hF : Measurable F) | |
| (hM : Measurable M) (hQ : Measurable Q) (hR : Measurable R) | |
| (hA : Measurable A) (hB : Measurable B) (hC : Measurable C) (hD : Measurable D) : | |
| Measurable (magneticContinuationThird ζ v F M Q R A B C D) := by | |
| have h1 := measurable_magneticContinuationSecond ζ v hF hM hQ hB hC hD | |
| have h2 := measurable_magneticContinuationSecond ζ v hF hM hQ (hA.mul hM) | |
| ((hB.mul hM).add (hA.mul hQ)) | |
| (((hC.mul hM).add (((measurable_const (a := (2 : ℝ))).mul hB).mul hQ)).add (hA.mul hR)) | |
| have h3 := measurable_magneticContinuationSecond ζ v hF hM hQ hA hB hC | |
| have h4 := measurable_magneticContinuationSecond ζ v hF hM hQ hM hQ hR | |
| have hTA := measurable_fieldSpinTransition ζ v hF hA | |
| have hTM := measurable_fieldSpinTransition ζ v hF hM | |
| have hSA := measurable_fieldTiltSpatial ζ v hF hM hA hB | |
| have hSM := measurable_fieldTiltSpatial ζ v hF hM hM hQ | |
| exact h1.add ((h2.sub (((h3.mul hTM).add | |
| ((measurable_const.mul hSA).mul hSM)).add (hTA.mul h4))).const_mul ζ) | |
| lemma magneticContinuationThird_bounded (ζ : ℝ) (v : ℝ≥0) | |
| {F M Q R A B C D : ℝ → ℝ} (hF : Measurable F) (hG : HasLinearGrowth F) | |
| (hM : MagneticContinuationBound M) (hQ : MagneticContinuationBound Q) | |
| (hR : MagneticContinuationBound R) (hA : MagneticContinuationBound A) | |
| (hB : MagneticContinuationBound B) (hC : MagneticContinuationBound C) | |
| (hD : MagneticContinuationBound D) : | |
| MagneticContinuationBound (magneticContinuationThird ζ v F M Q R A B C D) := by | |
| have T {A : ℝ → ℝ} (hA : MagneticContinuationBound A) := | |
| MagneticContinuationBound.transition ζ v hF hG hA | |
| have S {A B : ℝ → ℝ} (hA : MagneticContinuationBound A) (hB : MagneticContinuationBound B) := | |
| MagneticContinuationBound.spatial ζ v hF hG hM hA hB | |
| have D2 {A B C : ℝ → ℝ} (hA : MagneticContinuationBound A) | |
| (hB : MagneticContinuationBound B) (hC : MagneticContinuationBound C) := | |
| magneticContinuationSecond_bounded ζ v hF hG hM hQ hA hB hC | |
| have hb2 := (hB.mul hM).add (hA.mul hQ) | |
| have hb3 := ((hC.mul hM).add | |
| (((MagneticContinuationBound.const 2).mul hB).mul hQ)).add (hA.mul hR) | |
| exact (D2 hB hC hD).add ((MagneticContinuationBound.const ζ).mul | |
| ((D2 (hA.mul hM) hb2 hb3).sub | |
| ((((D2 hA hB hC).mul (T hM)).add | |
| ((((MagneticContinuationBound.const 2).mul (S hA hB)).mul (S hM hQ)))).add | |
| ((T hA).mul (D2 hM hQ hR))))) | |
| end InvariantIsing | |
| end | |
| end OAI | |