openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticContinuationComparison.lean
| 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 | |