| import OAI.Probability.InvariantIsing.Magnetic.MagneticFieldMinimum |
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| namespace OAI |
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| /-! The one-dimensional coercivity argument needed for the inverse mean |
| of a finite Gaussian slab. -/ |
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| noncomputable section |
| open Set Filter |
| open scoped Topology |
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| namespace InvariantIsing |
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| lemma exists_mean_root_of_linear_lower {F M : ℝ → ℝ} {C s : ℝ} |
| (dF : ∀ z, HasDerivAt F (M z) z) (hl : ∀ z, |z| - C ≤ F z) |
| (hs : |s| < 1) : ∃ z, M z = s := by |
| let δ := 1 - |s| |
| let R := (|F 0| + |C| + 1) / δ |
| have hδ : 0 < δ := sub_pos.mpr hs |
| have hR : 0 ≤ R := div_nonneg (by positivity) hδ.le |
| have he : δ * R = |F 0| + |C| + 1 := by |
| dsimp only [R] |
| field_simp |
| let G := fun z => F z - z * s |
| have hG : Continuous G := by |
| apply continuous_iff_continuousAt.mpr |
| intro z |
| exact ((dF z).sub ((hasDerivAt_id z).mul_const s)).continuousAt |
| have hlow (z : ℝ) : δ * |z| - C ≤ G z := by |
| have hz : z * s ≤ |z| * |s| := by |
| simpa only [abs_mul] using le_abs_self (z * s) |
| dsimp only [δ, G] |
| nlinarith [hl z] |
| obtain ⟨b, hb, hmin⟩ := isCompact_Icc.exists_isMinOn |
| (show (Icc (-R) R).Nonempty from ⟨0, by constructor <;> linarith⟩) hG.continuousOn |
| have hglobal (c : ℝ) : G b ≤ G c := by |
| by_cases hc : c ∈ Icc (-R) R |
| · exact hmin hc |
| have hcr : R ≤ |c| := by |
| by_contra hn |
| have hlt : |c| < R := lt_of_not_ge hn |
| exact hc ⟨(abs_lt.mp hlt).1.le, (abs_lt.mp hlt).2.le⟩ |
| have hlarge : F 0 ≤ G c := by |
| have hm := mul_le_mul_of_nonneg_left hcr hδ.le |
| have hcabs := le_abs_self C |
| have hfabs := le_abs_self (F 0) |
| linarith [hlow c] |
| have hz := hmin (show (0 : ℝ) ∈ Icc (-R) R by constructor <;> linarith) |
| change G b ≤ G 0 at hz |
| have hz' : G b ≤ F 0 := by |
| simpa only [G, zero_mul, sub_zero] using hz |
| exact hz'.trans hlarge |
| have hlocal : IsLocalMin G b := Eventually.of_forall hglobal |
| have hd : HasDerivAt G (M b - s) b := by |
| convert (dF b).sub ((hasDerivAt_id b).mul_const s) using 1 |
| · rfl |
| · simp only [one_mul] |
| exact ⟨b, sub_eq_zero.mp (hlocal.hasDerivAt_eq_zero hd)⟩ |
|
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| end InvariantIsing |
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| end |
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| end OAI |
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