openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Geometry /ChargedRequestSpace.lean
| import OAI.Combinatorics.EditApproximation.Geometry.ChargedGlobalQuery | |
| import OAI.Combinatorics.EditApproximation.Refinement.PhysicalRequestBits | |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryMemoSpace | |
| namespace OAI | |
| /-! A polynomial retained-cache bound for every execution. The estimate | |
| uses only fixed-width key insertion and computed parameter sizes, and | |
| therefore does not depend on an approximation success event. -/ | |
| namespace EditApproximation | |
| theorem computed_charged_request_bits_le (N nx ny passes copies : ℕ) (hN : 0 < N) | |
| (hell : 200 ≤ inputSmallLog N) (hnx : nx ≤ N) (hny : ny ≤ N) | |
| (hpasses : passes ≤ inputHeight N ^ 2) (hcopies : copies ≤ inputHeight N ^ 2) : | |
| ChargedPhysicalRequest.bits (integerParameters N).M | |
| (intervalTreeDepth (integerParameters N).M nx) ny passes copies | |
| (integerParameters N).T (integerParameters N).S ≤ 11 * inputHeight N + 3 := by | |
| let H := inputHeight N | |
| have hentry := computed_entryCapacity_bound N nx ny hN hell hnx hny | |
| have hbits := computed_request_bits_le N nx ny passes copies hN hell hnx hny hpasses hcopies | |
| have htable : PhysicalTableRequest.capacity (integerParameters N).M | |
| (intervalTreeDepth (integerParameters N).M nx) ny passes copies | |
| (integerParameters N).T (integerParameters N).S + 1 ≤ 2 ^ (11 * H + 1) := by | |
| exact (Nat.clog_le_iff_le_pow (by decide : 1 < (2 : ℕ))).mp hbits | |
| have hentry' : PhysicalTableRequest.entryCapacity (integerParameters N).M | |
| (intervalTreeDepth (integerParameters N).M nx) ny ≤ 2 ^ (11 * H + 1) := | |
| hentry.trans (pow_le_pow_right₀ (by decide : 1 ≤ (2 : ℕ)) (by omega)) | |
| unfold ChargedPhysicalRequest.bits | |
| apply (Nat.clog_le_iff_le_pow (by decide : 1 < (2 : ℕ))).2 | |
| unfold ChargedPhysicalRequest.capacity | |
| have hp : 2 ^ (11 * H + 3) = 4 * 2 ^ (11 * H + 1) := by | |
| rw [show 11 * H + 3 = (11 * H + 1) + 2 by omega, pow_add] | |
| norm_num | |
| omega | |
| change 2 * (_ + _) + 1 ≤ 2 ^ (11 * H + 3) | |
| rw [hp] | |
| omega | |
| theorem two_pow_inputHeight_le (N : ℕ) : 2 ^ inputHeight N ≤ 4 * (N + 2) ^ 2 := by | |
| have hc : 0 < Nat.clog 2 (N + 2) := | |
| Nat.clog_pos (by decide : 1 < (2 : ℕ)) (by omega) | |
| have hH := (rejectionWord_range (Nat.clog 2 (N + 2)) hc).2 | |
| have hN := (rejectionWord_range (N + 2) (by omega)).2 | |
| change inputHeight N < 2 * Nat.clog 2 (N + 2) at hH | |
| change 2 ^ Nat.clog 2 (N + 2) < 2 * (N + 2) at hN | |
| calc | |
| _ ≤ 2 ^ (2 * Nat.clog 2 (N + 2)) := pow_le_pow_right₀ (by decide : 1 ≤ (2 : ℕ)) hH.le | |
| _ = (2 ^ Nat.clog 2 (N + 2)) ^ 2 := by rw [← pow_mul]; congr 1; omega | |
| _ ≤ (2 * (N + 2)) ^ 2 := Nat.pow_le_pow_left hN.le 2 | |
| _ = _ := by ring | |
| theorem computed_charged_cache_nodes {κ : Type*} (N nx ny passes copies : ℕ) (hN : 0 < N) | |
| (hell : 200 ≤ inputSmallLog N) (hnx : nx ≤ N) (hny : ny ≤ N) | |
| (hpasses : passes ≤ inputHeight N ^ 2) (hcopies : copies ≤ inputHeight N ^ 2) | |
| (rank : κ → ℕ) (program : (key : κ) → FiniteQuery {child : κ // rank child < rank key}) | |
| (encode : κ → ℕ) (key : κ) : | |
| (rankedMemoEvaluate rank program | |
| (ChargedPhysicalRequest.bits (integerParameters N).M | |
| (intervalTreeDepth (integerParameters N).M nx) ny passes copies | |
| (integerParameters N).T (integerParameters N).S) encode key .empty).memory.nodes + 1 ≤ | |
| (N + 2) ^ 48 := by | |
| have hbits := computed_charged_request_bits_le N nx ny passes copies hN hell hnx hny hpasses hcopies | |
| have hcache := rankedMemoEvaluate_nodes_bound rank program | |
| (ChargedPhysicalRequest.bits (integerParameters N).M | |
| (intervalTreeDepth (integerParameters N).M nx) ny passes copies | |
| (integerParameters N).T (integerParameters N).S) encode key | |
| have hpow := Nat.pow_le_pow_right (by decide : 0 < (2 : ℕ)) (Nat.add_le_add_right hbits 1) | |
| apply (hcache.trans hpow).trans | |
| calc | |
| 2 ^ (11 * inputHeight N + 3 + 1) = 16 * (2 ^ inputHeight N) ^ 11 := by | |
| rw [show 11 * inputHeight N + 3 + 1 = inputHeight N * 11 + 4 by omega, pow_add, pow_mul] | |
| norm_num | |
| omega | |
| _ ≤ 16 * (4 * (N + 2) ^ 2) ^ 11 := Nat.mul_le_mul_left 16 (Nat.pow_le_pow_left (two_pow_inputHeight_le N) 11) | |
| _ = 2 ^ 26 * (N + 2) ^ 22 := by ring | |
| _ ≤ (N + 2) ^ 26 * (N + 2) ^ 22 := | |
| Nat.mul_le_mul_right _ (Nat.pow_le_pow_left (by omega) 26) | |
| _ = (N + 2) ^ 48 := by rw [← pow_add] | |
| end EditApproximation | |
| end OAI | |