| import OAI.Probability.InvariantIsing.Haar.HaarFrobeniusGeometry | |
| namespace OAI | |
| /-! Frobenius-Lipschitz observables are continuous in the matrix-group topology. -/ | |
| noncomputable section | |
| open Matrix Filter | |
| open scoped Topology | |
| namespace InvariantIsing | |
| lemma continuous_frobeniusDistance_right {N : ℕ} (U : SpecialOrthogonal N) : | |
| Continuous (fun V => frobeniusDistance V U) := by | |
| have he (i j : Fin N) : Continuous (fun V : SpecialOrthogonal N => | |
| (V : Matrix (Fin N) (Fin N) ℝ) i j) := | |
| (continuous_apply j).comp ((continuous_apply i).comp continuous_subtype_val) | |
| unfold frobeniusDistance | |
| apply Real.continuous_sqrt.comp | |
| exact continuous_finsetSum _ fun i _ => continuous_finsetSum _ fun j _ => | |
| ((he i j).sub continuous_const).pow 2 | |
| lemma frobeniusDistance_self {N : ℕ} (U : SpecialOrthogonal N) : frobeniusDistance U U = 0 := by | |
| simp [frobeniusDistance] | |
| theorem continuous_of_frobenius_lipschitz {N : ℕ} (f : SpecialOrthogonal N → ℝ) (L : ℝ) | |
| (hf : ∀ U V, |f U-f V| ≤ L*frobeniusDistance U V) : Continuous f := by | |
| apply continuous_iff_continuousAt.mpr | |
| intro U | |
| apply tendsto_iff_norm_sub_tendsto_zero.mpr | |
| apply squeeze_zero (fun V => norm_nonneg _) (fun V => by simpa only [Real.norm_eq_abs] using hf V U) | |
| have hh := ((continuous_frobeniusDistance_right U).tendsto U).const_mul L | |
| simpa only [frobeniusDistance_self,mul_zero] using hh | |
| end InvariantIsing | |
| end | |
| end OAI | |