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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 | import OAI.Probability.InvariantIsing.Gaussian.GaussianColumnDeletion
import OAI.Probability.InvariantIsing.Spectral.PositiveResolventBounds
namespace OAI
/-! Measurability of the actual deleted-column resolvent observables. -/
noncomputable section
open Matrix
open scoped BigOperators
namespace InvariantIsing
lemma continuous_gaussianGramLeaveOneOut {N m : ℕ} (j : Fin m) :
Continuous (gaussianGramLeaveOneOut (N := N) · j) := by
apply continuous_matrix
intro i k
simp only [gaussianGramLeaveOneOut,Matrix.smul_apply,Matrix.sum_apply,
Matrix.vecMulVec_apply,gaussianGramRawColumn,smul_eq_mul]
fun_prop
lemma continuous_gaussianGramLeaveResolvent {N m : ℕ} {t : ℝ} (ht : 0 < t) (j : Fin m) :
Continuous (gaussianGramLeaveResolvent (N := N) t · j) := by
apply continuous_iff_continuousAt.mpr
intro z
have hi := continuousAt_matrix_inv
(t • (1 : Matrix (Fin N) (Fin N) ℝ)+gaussianGramLeaveOneOut z j)
(show ContinuousAt Ring.inverse _ from by
simpa only [Ring.inverse_eq_inv'] using continuousAt_inv₀
(gaussianGramLeaveShift_posDef ht z j).det_pos.ne')
exact hi.comp (f := fun w : EuclideanSpace ℝ (Fin N × Fin m) =>
t • (1 : Matrix (Fin N) (Fin N) ℝ)+gaussianGramLeaveOneOut w j)
((continuous_const.add (continuous_gaussianGramLeaveOneOut j)).continuousAt)
lemma continuous_gaussianGramColumnQuadratic {N m : ℕ} {t : ℝ} (ht : 0 < t) (j : Fin m) :
Continuous (gaussianGramColumnQuadratic (N := N) t · j) := by
have hr := continuous_gaussianGramLeaveResolvent (N := N) ht j
simp only [gaussianGramColumnQuadratic,gaussianPatternScaledColumn,gaussianGramRawColumn,
Matrix.mulVec,dotProduct,Pi.smul_apply,smul_eq_mul]
fun_prop
lemma continuous_gaussianColumnArray {N m : ℕ} :
Continuous (gaussianColumnArray (N := N) (m := m)) := by
unfold gaussianColumnArray
exact (PiLp.continuous_toLp 2 (fun _ : Fin N × Fin m => ℝ)).comp
(continuous_pi (fun p => (continuous_apply p.1).comp (continuous_apply p.2)))
end InvariantIsing
end
end OAI
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