File size: 2,914 Bytes
d4317d7 | 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 66 67 | import OAI.MeasureTheory.Falconer.Recursion.PrefixPolicyGeometry
import Mathlib.Data.Set.Function
namespace OAI
namespace PlanarFalconer
theorem branch_charges_injective {n : ℕ} (start charge : ℕ → ℕ)
(C : Finset ℕ) (hC : C ⊆ Finset.range n)
(hstart : MonotoneOn start (Set.Icc 0 n))
(hcharge : ∀ j ∈ C, start j < charge j ∧ charge j ≤ start (j+1)) :
Set.InjOn charge C := by
intro i hi j hj he
have hiN := Finset.mem_range.mp (hC hi)
have hjN := Finset.mem_range.mp (hC hj)
have hci := hcharge i hi
have hcj := hcharge j hj
rcases lt_trichotomy i j with hij | hij | hji
· have hm := hstart (show i+1 ∈ Set.Icc 0 n by constructor <;> omega)
(show j ∈ Set.Icc 0 n by constructor <;> omega) (by omega : i+1 ≤ j)
omega
· exact hij
· have hm := hstart (show j+1 ∈ Set.Icc 0 n by constructor <;> omega)
(show i ∈ Set.Icc 0 n by constructor <;> omega) (by omega : j+1 ≤ i)
omega
theorem branch_representative_charges_bound (A : ℕ → ℝ) (N q n : ℕ)
(start charge : ℕ → ℕ) (C : Finset ℕ) (hC : C ⊆ Finset.range n)
(hstart : MonotoneOn start (Set.Icc 0 n))
(hcharge : ∀ j ∈ C, start j < charge j ∧ charge j ≤ start (j+1))
(hrep : ∀ j ∈ C, charge j ∈ policyRepresentatives A N q) :
C.card ≤ (policyRepresentatives A N q).card ∧ C.card ≤ N/q+1 := by
have hi := branch_charges_injective start charge C hC hstart hcharge
have hc : C.card ≤ (policyRepresentatives A N q).card := by
calc
_ = (C.image charge).card := (Finset.card_image_of_injOn hi).symm
_ ≤ _ := Finset.card_le_card (by
intro p hp
obtain ⟨j,hj,rfl⟩ := Finset.mem_image.mp hp
exact hrep j hj)
exact ⟨hc,hc.trans (policy_representatives_card A N q)⟩
theorem source_representatives_real_bound (A : ℕ → ℝ) {N : ℕ}
(hN : 0 < N) {kap : ℝ} (hkap : 0 < kap) (hkap1 : kap ≤ 1) :
((policyRepresentatives A N (sourcePolicyBlock N kap)).card:ℝ) ≤ 2/kap := by
let q := sourcePolicyBlock N kap
have hq : 0 < q := sourcePolicyBlock_pos hN hkap
have hqR : (0:ℝ) < q := Nat.cast_pos.mpr hq
have hc : ((policyRepresentatives A N q).card:ℝ) ≤ ((N/q:ℕ):ℝ)+1 := by
exact_mod_cast policy_representatives_card A N q
have hmul : ((N/q:ℕ):ℝ)*(q:ℝ) ≤ (N:ℝ) := by
exact_mod_cast Nat.div_mul_le_self N q
have hdiv : ((N/q:ℕ):ℝ) ≤ (N:ℝ)/(q:ℝ) := (le_div_iff₀ hqR).mpr hmul
have hblock := source_policy_block_lower N kap
have hrat : (N:ℝ)/(q:ℝ) ≤ 1/kap := (div_le_div_iff₀ hqR hkap).mpr (by
change (N:ℝ)*kap ≤ 1*(sourcePolicyBlock N kap:ℝ)
nlinarith)
have hone : 1 ≤ 1/kap := (le_div_iff₀ hkap).mpr (by simpa using hkap1)
change ((policyRepresentatives A N q).card:ℝ) ≤ 2/kap
have he : 2/kap = 1/kap+1/kap := by ring
rw [he]
linarith
end PlanarFalconer
end OAI
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