total_defect_lower_bound

formal source

 157          rw [← Finset.mul_sum]
 158
 159/-- Jensen lower bound: fixed log-charge implies a defect lower bound. -/
 160private theorem total_defect_lower_bound {N : ℕ} (hN : 0 < N) (c : Configuration N) :
 161    (N : ℝ) * Jlog (log_charge c / N) ≤ total_defect c := by
 162  have hN_pos : (0 : ℝ) < N := Nat.cast_pos.mpr hN
 163  have hw_nonneg : ∀ i ∈ (Finset.univ : Finset (Fin N)), 0 ≤ (1 / (N : ℝ)) := by
 164    intro _ _
 165    positivity
 166  have hw_sum : ∑ i ∈ (Finset.univ : Finset (Fin N)), (1 / (N : ℝ)) = 1 := by
 167    rw [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
 168    field_simp [hN_pos.ne']
 169  have hmem : ∀ i ∈ (Finset.univ : Finset (Fin N)), Real.log (c.entries i) ∈ (Set.univ : Set ℝ) := by
 170    intro _ _
 171    simp
 172  have hJensen :=
 173    Jlog_strictConvexOn.convexOn.map_sum_le
 174      (t := (Finset.univ : Finset (Fin N)))
 175      (w := fun _ : Fin N => (1 / (N : ℝ)))
 176      (p := fun i : Fin N => Real.log (c.entries i))
 177      hw_nonneg hw_sum hmem
 178  have hJensen' :
 179      Jlog (log_charge c / N) ≤ (1 / (N : ℝ)) * total_defect c := by
 180    have hJensen0 :
 181        Jlog (∑ i : Fin N, (1 / (N : ℝ)) * Real.log (c.entries i)) ≤
 182          ∑ i : Fin N, (1 / (N : ℝ)) * Jlog (Real.log (c.entries i)) := by
 183      simpa [smul_eq_mul] using hJensen
 184    rw [weighted_log_average hN c, weighted_Jlog_average c] at hJensen0
 185    exact hJensen0
 186  have hmul := mul_le_mul_of_nonneg_left hJensen' hN_pos.le
 187  simpa [div_eq_mul_inv, hN_pos.ne', mul_comm, mul_left_comm, mul_assoc] using hmul
 188
 189/-- Equality in the Jensen bound forces the configuration to be uniform. -/
 190private theorem eq_constant_config_of_defect_eq {N : ℕ} (hN : 0 < N) (c : Configuration N)