File size: 2,172 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 | import OAI.Probability.InvariantIsing.Magnetic.MagneticFieldMinimum
namespace OAI
/-! The one-dimensional coercivity argument needed for the inverse mean
of a finite Gaussian slab. -/
noncomputable section
open Set Filter
open scoped Topology
namespace InvariantIsing
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)⟩
end InvariantIsing
end
end OAI
|