| import OAI.Probability.InvariantIsing.Magnetic.MagneticPrefixComparison |
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| namespace OAI |
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| /-! The conditional square means preceding a transferred variance |
| increment decrease at fixed total mean spin. -/ |
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| noncomputable section |
| open Filter Set |
| open scoped NNReal Topology |
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| namespace InvariantIsing |
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| theorem magneticClosedSquare_adjacent_transfer (P L : List (ℝ × ℝ≥0)) |
| (hP : ∀ av ∈ P, 0 < av.1) (hP1 : ∀ av ∈ P, av.1 ≤ 1) |
| (hL : ∀ av ∈ L, 0 < av.1) (hL1 : ∀ av ∈ L, av.1 ≤ 1) |
| {a b : ℝ} (ha : 0 < a) (hab : a ≤ b) (hb1 : b ≤ 1) |
| (x δ y : ℝ≥0) |
| (hN : ∀ av ∈ P ++ (a, x + δ) :: (b, y) :: L, 0 < av.1) |
| (hO : ∀ av ∈ P ++ (a, x) :: (b, δ + y) :: L, 0 < av.1) |
| (i : ℕ) (hi : i ≤ P.length) {ζ v s : ℝ} |
| (hζ : 0 ≤ ζ) (hζ1 : ζ ≤ 1) (hv : 0 ≤ v) (hs : s ∈ Icc (-1 : ℝ) 1) : |
| closedMagneticContinuation (P ++ (a, x + δ) :: (b, y) :: L) |
| (magneticScalarSquareFourJet _ hN |
| ⟨i, by simp only [List.length_append, List.length_cons]; omega⟩).toMagneticContinuationJet |
| ζ (v, s) ≤ |
| closedMagneticContinuation (P ++ (a, x) :: (b, δ + y) :: L) |
| (magneticScalarSquareFourJet _ hO |
| ⟨i, by simp only [List.length_append, List.length_cons]; omega⟩).toMagneticContinuationJet |
| ζ (v, s) := by |
| let N := (a, x + δ) :: (b, y) :: L |
| let O := (a, x) :: (b, δ + y) :: L |
| have hNT : ∀ av ∈ N, 0 < av.1 := fun av hav => hN av (List.mem_append_right P hav) |
| have hOT : ∀ av ∈ O, 0 < av.1 := fun av hav => hO av (List.mem_append_right P hav) |
| have hNT1 : ∀ av ∈ N, av.1 ≤ 1 := by |
| intro av hav |
| rcases List.mem_cons.mp hav with rfl | hav |
| · exact hab.trans hb1 |
| · rcases List.mem_cons.mp hav with rfl | hav |
| · exact hb1 |
| · exact hL1 av hav |
| have hOT1 : ∀ av ∈ O, av.1 ≤ 1 := by |
| intro av hav |
| rcases List.mem_cons.mp hav with rfl | hav |
| · exact hab.trans hb1 |
| · rcases List.mem_cons.mp hav with rfl | hav |
| · exact hb1 |
| · exact hL1 av hav |
| have hinit (u : ℝ) (hu : u ∈ Icc (-1 : ℝ) 1) : |
| closedMagneticScalarCurvature N hNT 0 (0, u) ≤ |
| closedMagneticScalarCurvature O hOT 0 (0, u) := |
| magneticClosedCurvature_adjacent_transfer L hL hL1 ha hab hb1 x δ y |
| (le_refl 0) (by norm_num) (le_refl 0) hu |
| exact magneticClosedSquare_append_mono P N O hP hP1 hNT hNT1 hOT hOT1 hinit i hi |
| hζ hζ1 hv hs |
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| end InvariantIsing |
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| end |
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| end OAI |
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