File size: 3,058 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 | import OAI.Probability.InvariantIsing.Magnetic.MagneticParabolicComparison
namespace OAI
/-! Increasing the inverse-curvature coefficient increases a convex
continued observable. This is the comparison used for conditional square means. -/
noncomputable section
open Filter Set
open scoped Topology
namespace InvariantIsing
theorem magnetic_continuation_coefficient_comparison {T : ℝ} (hT : 0 ≤ T)
(U Ut Ux Uxx V Vt Vx Vxx a b : ℝ × ℝ → ℝ)
(hU : ContinuousOn U (Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1))
(hV : ContinuousOn V (Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1))
(hinit : ∀ s ∈ Icc (-1 : ℝ) 1, U (0, s) ≤ V (0, s))
(hUt : ∀ t ∈ Ioc (0 : ℝ) T, ∀ s ∈ Icc (-1 : ℝ) 1,
HasDerivWithinAt (fun r => U (r, s)) (Ut (t, s)) (Iic t) t)
(hVt : ∀ t ∈ Ioc (0 : ℝ) T, ∀ s ∈ Icc (-1 : ℝ) 1,
HasDerivWithinAt (fun r => V (r, s)) (Vt (t, s)) (Iic t) t)
(hUx : ∀ t ∈ Icc (0 : ℝ) T, ∀ s ∈ Ioo (-1 : ℝ) 1,
HasDerivAt (fun r => U (t, r)) (Ux (t, s)) s)
(hVx : ∀ t ∈ Icc (0 : ℝ) T, ∀ s ∈ Ioo (-1 : ℝ) 1,
HasDerivAt (fun r => V (t, r)) (Vx (t, s)) s)
(hUxx : ∀ t ∈ Icc (0 : ℝ) T, ∀ s ∈ Ioo (-1 : ℝ) 1,
HasDerivAt (fun r => Ux (t, r)) (Uxx (t, s)) s)
(hVxx : ∀ t ∈ Icc (0 : ℝ) T, ∀ s ∈ Ioo (-1 : ℝ) 1,
HasDerivAt (fun r => Vx (t, r)) (Vxx (t, s)) s)
(ha : ∀ p ∈ Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, 0 ≤ a p)
(hab : ∀ p ∈ Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, a p ≤ b p)
(hconvex : ∀ p ∈ Ioc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, 0 ≤ Uxx p)
(hbdy : ∀ t ∈ Icc (0 : ℝ) T, b (t, -1) = 0 ∧ b (t, 1) = 0)
(hPU : ∀ p ∈ Ioc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, Ut p = a p ^ 2 / 2 * Uxx p)
(hPV : ∀ p ∈ Ioc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, Vt p = b p ^ 2 / 2 * Vxx p) :
∀ p ∈ Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1, U p ≤ V p := by
have hn := parabolic_minimum_nonneg_bounded (M := 0) hT
(fun p => V p - U p) (fun p => Vt p - Ut p)
(fun p => Vx p - Ux p) (fun p => Vxx p - Uxx p)
(fun p => b p ^ 2 / 2) (fun _ => 0) (fun _ => 0)
(hV.sub hU) (fun s hs => sub_nonneg.mpr (hinit s hs))
(fun t ht s hs => (hVt t ht s hs).sub (hUt t ht s hs))
(fun t ht s hs => (hVx t ht s hs).sub (hUx t ht s hs))
(fun t ht s hs => (hVxx t ht s hs).sub (hUxx t ht s hs))
(fun p hp => div_nonneg (sq_nonneg _) (by norm_num))
(fun p hp => le_refl 0)
(fun t ht => by
rcases hbdy t ht with ⟨hl, hr⟩
simp only [hl, hr, zero_pow (by norm_num : 2 ≠ 0), zero_div, and_self]) ?_
· intro p hp
exact sub_nonneg.mp (hn p hp)
· intro p hp
have hp' : p ∈ Icc (0 : ℝ) T ×ˢ Icc (-1 : ℝ) 1 := ⟨⟨hp.1.1.le, hp.1.2⟩, hp.2⟩
have hs : 0 ≤ (b p) ^ 2 - (a p) ^ 2 := by
have ha0 := ha p hp'
have hab' := hab p hp'
nlinarith
have hmul := mul_nonneg hs (hconvex p hp)
rw [hPV p hp, hPU p hp]
simp only [zero_mul, add_zero]
nlinarith
end InvariantIsing
end
end OAI
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