openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Geometry /ActualRepresentatives.lean
| import OAI.Combinatorics.EditApproximation.Geometry.InitialConeGrid | |
| namespace OAI | |
| /-! The finite representative list and the actual rounded comparison representative. -/ | |
| namespace EditApproximation | |
| def representativeList {n : ℕ} (N P F : ℕ) (a : ℚ) | |
| (initial : TargetInterval n → ℚ) : List (ℕ × TargetInterval n) := | |
| (dyadicScales N).flatMap fun b => | |
| ((targetStates n).filter fun r => decide | |
| (seedGridSpacing b P ∣ r.lo ∧ seedGridSpacing b P ∣ r.hi ∧ | |
| (b : ℚ) / (4 * a) ≤ initial r ∧ initial r ≤ 8 * F * b)).map fun r => (b, r) | |
| theorem representativeList_mem {n N P F b : ℕ} (a : ℚ) | |
| (initial : TargetInterval n → ℚ) (r : TargetInterval n) | |
| (hb : b ∈ dyadicScales N) | |
| (hgrid : seedGridSpacing b P ∣ r.lo ∧ seedGridSpacing b P ∣ r.hi) | |
| (hlower : (b : ℚ) / (4 * a) ≤ initial r) (hupper : initial r ≤ 8 * F * b) : | |
| (b, r) ∈ representativeList N P F a initial := by | |
| apply List.mem_flatMap.mpr | |
| refine ⟨b, hb, List.mem_map.mpr ⟨r, List.mem_filter.mpr ⟨targetStates_complete n r, ?_⟩, rfl⟩⟩ | |
| simpa using ⟨hgrid.1, hgrid.2, hlower, hupper⟩ | |
| theorem rounded_comparison_representative {α : Type*} (source target : List α) | |
| (N P F : ℕ) (a : ℚ) (hP : 4 ≤ P) (hF : 0 ≤ (F : ℚ)) (ha : 0 < a) | |
| (initial : TargetInterval target.length → ℚ) | |
| (hlower : ∀ r, (stateEditCost source target r : ℚ) / a ≤ initial r) | |
| (hupper : ∀ r, initial r ≤ 2 * F * stateEditCost source target r) | |
| (q : TargetInterval target.length) (hc : 0 < stateEditCost source target q) | |
| (hN : stateEditCost source target q ≤ N) : | |
| let b := 2 ^ Nat.log 2 (stateEditCost source target q) | |
| (b, roundInterval (seedGridSpacing b P) q) ∈ representativeList N P F a initial := by | |
| dsimp only | |
| let b := 2 ^ Nat.log 2 (stateEditCost source target q) | |
| let r := roundInterval (seedGridSpacing b P) q | |
| have hscale := cost_dyadicScale _ N hc hN | |
| have hscaleLower : (b : ℚ) ≤ stateEditCost source target q := by exact_mod_cast hscale.2.1 | |
| have hscaleUpper : (stateEditCost source target q : ℚ) < 2 * b := by exact_mod_cast hscale.2.2 | |
| have he := rounded_seedGrid_error b P (by omega) q | |
| have he0 : (0 : ℚ) ≤ targetEndpointDistance q r := Nat.cast_nonneg _ | |
| have hPr : (4 : ℚ) ≤ P := by exact_mod_cast hP | |
| have hsmall : (targetEndpointDistance q r : ℚ) ≤ b / 2 := by nlinarith only [he, he0, hPr] | |
| have hdist := abs_le.mp (stateEditCost_distance_abs source target q r) | |
| have hcrlo : (b : ℚ) / 4 ≤ stateEditCost source target r := by linarith only [hdist.1, hdist.2, hsmall, hscaleLower] | |
| have hcrhi : (stateEditCost source target r : ℚ) ≤ 4 * b := by | |
| have hb0 : (0 : ℚ) ≤ b := Nat.cast_nonneg _ | |
| linarith only [hdist.1, hdist.2, hsmall, hscaleUpper, hb0] | |
| apply representativeList_mem a initial r hscale.1 (roundInterval_on_grid _ q) | |
| · have hdiv := div_le_div_of_nonneg_right hcrlo ha.le | |
| have hid : (b : ℚ) / 4 / a = b / (4 * a) := by ring | |
| rw [hid] at hdiv | |
| exact hdiv.trans (hlower r) | |
| · have hmul := mul_le_mul_of_nonneg_left hcrhi (mul_nonneg (by norm_num : (0 : ℚ) ≤ 2) hF) | |
| exact (hupper r).trans (by nlinarith only [hmul]) | |
| end EditApproximation | |
| end OAI | |