openAI-math-dataset / lean /OAI /MeasureTheory /Falconer /Recursion /BranchRepresentativeCharges.lean
| 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 | |