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f0f05de | 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 | import Mathlib
namespace OAI
/-!
# Deficit charges in the finite-tree separation induction
The exact deficit charges are those of `prop:finite-separation`.
-/
namespace TreeEdit.BinaryLower.FiniteTree
/-- At least half the children each charge `q` unmatched symbols. -/
theorem row_deficit_charge {m : ℕ} (bad : Finset (Fin m))
(δ : Fin m → ℝ) (q : ℝ) (hq : 0 ≤ q)
(hnonneg : ∀ i, 0 ≤ δ i) (hbad : ∀ i ∈ bad, q ≤ δ i)
(hhalf : m ≤ 2 * bad.card) :
(m : ℝ) * q / 2 ≤ ∑ i, δ i := by
classical
have hpoint (i : Fin m) : (if i ∈ bad then q else 0) ≤ δ i := by
split_ifs with hi
· exact hbad i hi
· exact hnonneg i
have hsum := Finset.sum_le_sum (fun i (_ : i ∈ Finset.univ) => hpoint i)
have hcalc : (∑ i : Fin m, if i ∈ bad then q else 0) = (bad.card : ℝ) * q := by
simp
rw [hcalc] at hsum
have hmR : (m : ℝ) ≤ 2 * (bad.card : ℝ) := by exact_mod_cast hhalf
nlinarith [mul_le_mul_of_nonneg_right hmR hq]
/-- The chosen row charge is no larger than the tagged transition gap. -/
theorem finite_charge_le_gap (m h : ℕ) (hm : 2 ≤ m) (B : ℝ) (hB : 0 ≤ B) :
(m : ℝ) * (B / (4 : ℝ) ^ h) / 2 ≤ ((m : ℝ) - 1) * B := by
have hp : (1 : ℝ) ≤ (4 : ℝ) ^ h := one_le_pow₀ (by norm_num)
have hp0 : 0 < (4 : ℝ) ^ h := by positivity
have hq : B / (4 : ℝ) ^ h ≤ B := by
apply (div_le_iff₀ hp0).2
nlinarith
have hmR : (2 : ℝ) ≤ m := by exact_mod_cast hm
have hm0 : (0 : ℝ) ≤ m := Nat.cast_nonneg _
nlinarith [mul_le_mul_of_nonneg_left hq hm0]
/-- Half the total row charge is exactly the next `4⁻ʰ` separation scale. -/
theorem finite_charge_identity (m T h : ℕ) (B : ℝ) :
(T : ℝ) * ((m : ℝ) * (B / (4 : ℝ) ^ h) / 2) / 2 =
((m : ℝ) * T * B) / (4 : ℝ) ^ (h + 1) := by
rw [pow_succ]
ring
end TreeEdit.BinaryLower.FiniteTree
end OAI
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