openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticBoundedInverseComparison.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticParabolicNegative | |
| namespace OAI | |
| /-! Exponent comparison for the one-dimensional inverse-curvature equation with an arbitrary finite curvature cap. | |
| The coefficient bound is required only where the proposed order fails. -/ | |
| noncomputable section | |
| open Filter Set | |
| open scoped Topology | |
| namespace InvariantIsing | |
| theorem inverse_curvature_parabolic_comparison_bound {T ζ η K A : ℝ} | |
| (hT : 0 ≤ T) (hη : 0 ≤ η) (hζη : ζ ≤ η) | |
| (a aTime ax axx b bTime bx bxx : ℝ × ℝ → ℝ) | |
| (ha : ContinuousOn a (Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1)) | |
| (hb : ContinuousOn b (Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1)) | |
| (hinit : ∀ s ∈ Icc (-1 : ℝ) 1, a (0, s) ≤ b (0, s)) | |
| (hat : ∀ t ∈ Ioc (0 : ℝ) T, ∀ s ∈ Icc (-1 : ℝ) 1, | |
| HasDerivWithinAt (fun u => a (u, s)) (aTime (t, s)) (Iic t) t) | |
| (hbt : ∀ t ∈ Ioc (0 : ℝ) T, ∀ s ∈ Icc (-1 : ℝ) 1, | |
| HasDerivWithinAt (fun u => b (u, s)) (bTime (t, s)) (Iic t) t) | |
| (hax : ∀ t ∈ Icc (0 : ℝ) T, ∀ s ∈ Ioo (-1 : ℝ) 1, | |
| HasDerivAt (fun u => a (t, u)) (ax (t, s)) s) | |
| (hbx : ∀ t ∈ Icc (0 : ℝ) T, ∀ s ∈ Ioo (-1 : ℝ) 1, | |
| HasDerivAt (fun u => b (t, u)) (bx (t, s)) s) | |
| (haxx : ∀ t ∈ Icc (0 : ℝ) T, ∀ s ∈ Ioo (-1 : ℝ) 1, | |
| HasDerivAt (fun u => ax (t, u)) (axx (t, s)) s) | |
| (hbxx : ∀ t ∈ Icc (0 : ℝ) T, ∀ s ∈ Ioo (-1 : ℝ) 1, | |
| HasDerivAt (fun u => bx (t, u)) (bxx (t, s)) s) | |
| (ha0 : ∀ p ∈ Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, 0 ≤ a p) | |
| (hb0 : ∀ p ∈ Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, 0 ≤ b p) | |
| (ha1 : ∀ p ∈ Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, a p ≤ A) | |
| (hbound : ∀ p ∈ Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, |a p * axx p| ≤ K) | |
| (hend : ∀ t ∈ Icc (0 : ℝ) T, a (t, -1) = 0 ∧ b (t, -1) = 0 ∧ | |
| a (t, 1) = 0 ∧ b (t, 1) = 0) | |
| (hpa : ∀ p ∈ Ioc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, | |
| aTime p = a p ^ 2 / 2 * axx p + ζ * a p ^ 2) | |
| (hpb : ∀ p ∈ Ioc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, | |
| bTime p = b p ^ 2 / 2 * bxx p + η * b p ^ 2) : | |
| ∀ p ∈ Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, a p ≤ b p := by | |
| have h := parabolic_minimum_nonneg_bounded_on_negative (M := K + 2 * A * η) hT | |
| (fun p => b p - a p) (fun p => bTime p - aTime p) (fun p => bx p - ax p) | |
| (fun p => bxx p - axx p) (fun p => b p ^ 2 / 2) (fun _ => 0) | |
| (fun p => (b p + a p) * (axx p / 2 + η)) | |
| (hb.sub ha) (fun s hs => sub_nonneg.mpr (hinit s hs)) | |
| (fun t ht s hs => (hbt t ht s hs).sub (hat t ht s hs)) | |
| (fun t ht s hs => (hbx t ht s hs).sub (hax t ht s hs)) | |
| (fun t ht s hs => (hbxx t ht s hs).sub (haxx t ht s hs)) | |
| (fun p hp => div_nonneg (sq_nonneg _) (by norm_num)) | |
| (fun p hp hn => by | |
| exact inverse_curvature_difference_coefficient_bound | |
| (ha0 p hp) (hb0 p hp) (by linarith) (ha1 p hp) hη (hbound p hp)) | |
| (fun t ht => by | |
| obtain ⟨haL, hbL, haR, hbR⟩ := hend t ht | |
| simp only [hbL, hbR, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, | |
| zero_div, and_self]) ?_ | |
| · intro p hp | |
| exact sub_nonneg.mp (h p hp) | |
| · intro p hp | |
| rw [hpb p hp, hpa p hp] | |
| have hn := mul_nonneg (sub_nonneg.mpr hζη) (sq_nonneg (a p)) | |
| nlinarith | |
| end InvariantIsing | |
| end | |
| end OAI | |