openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Refinement /ComputedCurrentQueryBudget.lean
| import OAI.Combinatorics.EditApproximation.Geometry.CurrentQueryInputs | |
| import OAI.Combinatorics.EditApproximation.Refinement.ComputedGroupNumerical | |
| namespace OAI | |
| /-! Explicit polynomial same-index call budget after substituting the | |
| computed window, cell, scale-label, sampling and rounding parameters. -/ | |
| namespace EditApproximation | |
| def localCurrentBudget (window cells labels M Q inverseTheta : ℕ) : ℕ := | |
| window * cells * labels * (624 * M * Q * inverseTheta ^ 2) + window * cells * M + window * M | |
| theorem localCurrentBudget_poly (ell M window cells labels : ℕ) (hell : 4 ≤ ell) | |
| (hwindow : window ≤ 2 * ell ^ 2) (hcells : cells ≤ 1296 * ell ^ 82) | |
| (hlabels : labels ≤ 2 * ell) : | |
| localCurrentBudget window cells labels M (ell ^ 80) (ell ^ 40) ≤ M * ell ^ 256 := by | |
| have hfirst := Nat.mul_le_mul | |
| (Nat.mul_le_mul (Nat.mul_le_mul hwindow hcells) hlabels) | |
| (le_rfl (a := 624 * M * ell ^ 80 * (ell ^ 40) ^ 2)) | |
| have hsecond := Nat.mul_le_mul_right M (Nat.mul_le_mul hwindow hcells) | |
| have hthird := Nat.mul_le_mul_right M hwindow | |
| have h84 : ell ^ 84 ≤ ell ^ 245 := pow_le_pow_right₀ (by omega : 1 ≤ ell) (by decide : 84 ≤ 245) | |
| have h2 : ell ^ 2 ≤ ell ^ 245 := pow_le_pow_right₀ (by omega : 1 ≤ ell) (by decide : 2 ≤ 245) | |
| have hsecond' : 2 * ell ^ 2 * (1296 * ell ^ 82) * M ≤ 2592 * M * ell ^ 245 := by | |
| calc | |
| _ = 2592 * M * ell ^ 84 := by ring | |
| _ ≤ _ := Nat.mul_le_mul_left _ h84 | |
| have hthird' : 2 * ell ^ 2 * M ≤ 2 * M * ell ^ 245 := by | |
| calc | |
| _ = 2 * M * ell ^ 2 := by ring | |
| _ ≤ _ := Nat.mul_le_mul_left _ h2 | |
| have hsum : localCurrentBudget window cells labels M (ell ^ 80) (ell ^ 40) ≤ | |
| 3237410 * M * ell ^ 245 := by | |
| apply (Nat.add_le_add (Nat.add_le_add hfirst (hsecond.trans hsecond')) | |
| (hthird.trans hthird')).trans_eq | |
| ring | |
| have hconstant : 3237410 ≤ ell ^ 11 := by | |
| apply (show 3237410 ≤ (4 : ℕ) ^ 11 by norm_num).trans | |
| exact Nat.pow_le_pow_left hell 11 | |
| calc | |
| _ ≤ 3237410 * M * ell ^ 245 := hsum | |
| _ ≤ ell ^ 11 * M * ell ^ 245 := Nat.mul_le_mul_right _ (Nat.mul_le_mul_right M hconstant) | |
| _ = M * ell ^ 256 := by ring | |
| theorem computed_window_exponent_bound (N : ℕ) (hN : 0 < N) (hell : 8 ≤ inputSmallLog N) : | |
| computedSeedInitialExponent N + 8 ≤ 2 * inputSmallLog N ^ 2 := by | |
| have hh := (inputSmallLog_bounds N hN).1 | |
| have hp : coarseExponent (inputSmallLog N) ≤ inputSmallLog N := | |
| (Nat.sqrt_le_self _).trans ((Nat.sqrt_le_self _).trans (Nat.log_le_self _ _)) | |
| have hmul := Nat.mul_le_mul (Nat.add_le_add_right hp 4) hh | |
| unfold computedSeedInitialExponent | |
| have hslack : 4 * inputSmallLog N + 8 ≤ inputSmallLog N ^ 2 := by nlinarith | |
| nlinarith only [hmul, hslack] | |
| theorem computed_local_cell_budget (N : ℕ) (hell : 1 ≤ inputSmallLog N) : | |
| ((32 * inputSmallLog N + 2) * 2 ^ (40 * smallLogExponent N) + 2) ^ 2 ≤ | |
| 1296 * inputSmallLog N ^ 82 := by | |
| have hpower : 2 ^ (40 * smallLogExponent N) = inputSmallLog N ^ 40 := by | |
| unfold inputSmallLog | |
| rw [← pow_mul, Nat.mul_comm] | |
| rw [hpower] | |
| have hinner : (32 * inputSmallLog N + 2) * inputSmallLog N ^ 40 + 2 ≤ | |
| 36 * inputSmallLog N ^ 41 := by | |
| have ha := Nat.mul_le_mul_right (inputSmallLog N ^ 40) (show 32 * inputSmallLog N + 2 ≤ | |
| 34 * inputSmallLog N by omega) | |
| have hb : 2 ≤ 2 * inputSmallLog N ^ 41 := Nat.mul_le_mul_left 2 (one_le_pow₀ hell) | |
| calc | |
| _ ≤ 34 * inputSmallLog N * inputSmallLog N ^ 40 + 2 := Nat.add_le_add_right ha _ | |
| _ ≤ 34 * inputSmallLog N * inputSmallLog N ^ 40 + 2 * inputSmallLog N ^ 41 := | |
| Nat.add_le_add_left hb _ | |
| _ = _ := by ring | |
| calc | |
| _ ≤ (36 * inputSmallLog N ^ 41) ^ 2 := Nat.pow_le_pow_left hinner 2 | |
| _ = _ := by ring | |
| theorem computed_refinement_window (N : ℕ) (hN : 0 < N) (hell : 8 ≤ inputSmallLog N) | |
| (A w : ℚ) (hA : A ≤ ((integerParameters N).initialFactor : ℚ)) : | |
| (dyadicQueryWindow N w (refinementFactor A (inputSmallLog N)) (inputSmallLog N)).length ≤ | |
| computedSeedInitialExponent N + 8 := by | |
| have hh := inputSmallLog_bounds N hN | |
| have hf : smallLogExponent N ≤ inputSmallLog N := (smallLogExponent N).lt_two_pow_self.le | |
| have he0 : smallLogExponent N ≤ computedSeedInitialExponent N := by | |
| have hfour := Nat.mul_le_mul_right (heightExponent N) | |
| (show 4 ≤ coarseExponent (inputSmallLog N) + 4 by omega) | |
| unfold computedSeedInitialExponent | |
| nlinarith only [hh.2, hf, hfour] | |
| apply refinement_query_window_length N (inputSmallLog N) (computedSeedInitialExponent N) A w (by omega) | |
| · simpa only [inputSmallLog, Nat.cast_pow, Nat.cast_ofNat] using | |
| pow_le_pow_right₀ (by norm_num : (1 : ℚ) ≤ 2) he0 | |
| · exact hA.trans_eq (computedSeed_initial_factor N).symm | |
| /-- The actual finite local shared/selected/online input cover is bounded | |
| by M*ell^256 on every seed. The online-loop length premise is the literal | |
| one-action-per-applicable-scale property proved for the physical program. -/ | |
| theorem computed_currentQueryInputCover_card {α : Type*} (source target : List α) | |
| (N P : ℕ) (hN : 0 < N) (hell : 200 ≤ inputSmallLog N) | |
| (parent : TargetInterval source.length) (A tau : ℚ) | |
| (hA : A ≤ ((integerParameters N).initialFactor : ℚ)) | |
| (parentInitial : TargetInterval target.length → ℚ) | |
| (child initial current : Fin (integerParameters N).M → TargetInterval target.length → ℚ) | |
| (draw : CellGroupDraws (integerParameters N).M (inputSmallLog N) (inputSmallLog N ^ 80)) | |
| (q : TargetInterval target.length) (hw : 0 ≤ parentInitial q) | |
| (online : List (PhysicalRationalAction (integerParameters N).M target.length)) | |
| (honline : online.length ≤ | |
| (dyadicQueryWindow N (parentInitial q) (refinementFactor A (inputSmallLog N)) (inputSmallLog N)).length) : | |
| (currentQueryInputCover source target parent N P (inputSmallLog N) (40 * smallLogExponent N) | |
| (inputSmallLog N ^ 80) tau (refinementFactor A (inputSmallLog N)) parentInitial | |
| child initial current draw q online).card ≤ (integerParameters N).M * inputSmallLog N ^ 256 := by | |
| let ell := inputSmallLog N | |
| let M := (integerParameters N).M | |
| let W := computedSeedInitialExponent N + 8 | |
| let C := ((32 * ell + 2) * 2 ^ (40 * smallLogExponent N) + 2) ^ 2 | |
| have hell8 : 8 ≤ ell := by omega | |
| have hM : 0 < M := by have h := (computed_branching_bounds N hN hell).1; omega | |
| have hlabels (b : ℕ) (hb : b ∈ dyadicScales N) : (groupDyadicScales M b ell).length ≤ 2 * ell := by | |
| obtain ⟨eb, _, rfl⟩ := List.mem_map.mp hb | |
| exact computed_group_scale_count N eb hN hell8 | |
| have hwindow := computed_refinement_window N hN hell8 A (parentInitial q) hA | |
| have hkeys : (localGroupQueryKeys N ell (40 * smallLogExponent N) | |
| (refinementFactor A ell) parentInitial q).length ≤ W * C := | |
| (localGroupQueryKeys_length N ell (40 * smallLogExponent N) (refinementFactor A ell) | |
| parentInitial q (by omega) hw).trans (Nat.mul_le_mul_right C hwindow) | |
| have hgroups := localCellGroupActions_length_refinement source target parent N P ell | |
| (40 * smallLogExponent N) (ell ^ 80) (computedSeedInitialExponent N) tau A | |
| parentInitial child initial current draw q (by omega) hw | |
| have hraw := currentQueryInputCover_card source target parent N P ell (40 * smallLogExponent N) | |
| (ell ^ 80) (2 * ell) tau (refinementFactor A ell) parentInitial child initial current draw q online | |
| hM (by omega) (one_le_pow₀ (by omega)) hlabels | |
| have hrawN : (currentQueryInputCover source target parent N P ell (40 * smallLogExponent N) | |
| (ell ^ 80) tau (refinementFactor A ell) parentInitial child initial current draw q online).card ≤ | |
| (localGroupQueryKeys N ell (40 * smallLogExponent N) (refinementFactor A ell) parentInitial q).length * | |
| (2 * ell) * (624 * M * ell ^ 80 * (2 ^ (40 * smallLogExponent N)) ^ 2) + | |
| (localCellGroupActions source target parent N P ell (40 * smallLogExponent N) (ell ^ 80) tau | |
| (refinementFactor A ell) parentInitial child initial current draw q).length * M + online.length * M := by | |
| exact_mod_cast hraw | |
| have hFpower : (ell : ℚ) ≤ (2 : ℚ) ^ computedSeedInitialExponent N := by | |
| have hh := inputSmallLog_bounds N hN | |
| have hf : smallLogExponent N ≤ ell := (smallLogExponent N).lt_two_pow_self.le | |
| have hfour := Nat.mul_le_mul_right (heightExponent N) | |
| (show 4 ≤ coarseExponent ell + 4 by omega) | |
| have he : smallLogExponent N ≤ computedSeedInitialExponent N := by | |
| unfold computedSeedInitialExponent | |
| nlinarith only [hh.2, hf, hfour] | |
| simpa only [ell, inputSmallLog, Nat.cast_pow, Nat.cast_ofNat] using | |
| pow_le_pow_right₀ (by norm_num : (1 : ℚ) ≤ 2) he | |
| have hAbound : A ≤ (2 : ℚ) ^ computedSeedInitialExponent N := hA.trans_eq (computedSeed_initial_factor N).symm | |
| have hg : (localCellGroupActions source target parent N P ell (40 * smallLogExponent N) (ell ^ 80) tau | |
| (refinementFactor A ell) parentInitial child initial current draw q).length ≤ W * C := | |
| hgroups hFpower hAbound | |
| have hbudget : (currentQueryInputCover source target parent N P ell (40 * smallLogExponent N) | |
| (ell ^ 80) tau (refinementFactor A ell) parentInitial child initial current draw q online).card ≤ | |
| localCurrentBudget W C (2 * ell) M (ell ^ 80) (2 ^ (40 * smallLogExponent N)) := by | |
| exact hrawN.trans (Nat.add_le_add (Nat.add_le_add | |
| (Nat.mul_le_mul_right _ (Nat.mul_le_mul_right (2 * ell) hkeys)) | |
| (Nat.mul_le_mul_right M hg)) (Nat.mul_le_mul_right M (honline.trans hwindow))) | |
| have hpower : 2 ^ (40 * smallLogExponent N) = ell ^ 40 := by | |
| dsimp only [ell] | |
| unfold inputSmallLog | |
| rw [← pow_mul, Nat.mul_comm] | |
| rw [hpower] at hbudget | |
| exact hbudget.trans (localCurrentBudget_poly ell M W C (2 * ell) | |
| ((by decide : 4 ≤ (8 : ℕ)).trans hell8) | |
| (computed_window_exponent_bound N hN hell8) | |
| (computed_local_cell_budget N ((by decide : 1 ≤ (8 : ℕ)).trans hell8)) le_rfl) | |
| end EditApproximation | |
| end OAI | |