File size: 3,316 Bytes
a89d282 | 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 | 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
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