openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Separators /SeparatorSamplerBounds.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Separators.SeparatorSamplerCells | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Separators.SeparatorStoredSpace | |
| namespace OAI | |
| /-! # The concrete family-sampler storage under the capacity guard -/ | |
| namespace LowSpaceSubsetSum.SeparatorSamplerBounds | |
| open RepresentationSetup | |
| private theorem reserve_le (n g u : ℕ) (hn : n+1 ≤ u) (hg : g ≤ n) (hu : 10000 ≤ u) : | |
| FixedSizeSampleCells.reserve n g+18 ≤ u^5 := by | |
| have h3 := Nat.pow_le_pow_left (show n ≤ u by omega) 3 | |
| have hg3 := Nat.pow_le_pow_left (show g+1 ≤ u by omega) 3 | |
| have hu3 : u ≤ u^3 := by | |
| simpa only [pow_one] using pow_le_pow_right' (by omega : 1 ≤ u) (by decide : 1 ≤ 3) | |
| have hu4 : u ≤ u^4 := by | |
| simpa only [pow_one] using pow_le_pow_right' (by omega : 1 ≤ u) (by decide : 1 ≤ 4) | |
| have h34 : u^3 ≤ u^4 := pow_le_pow_right' (by omega) (by decide) | |
| have ht : 3*n^3+128 ≤ 4*u^3 := by nlinarith only [h3, hu3, hu] | |
| have hm := Nat.mul_le_mul (show g+1 ≤ u by omega) ht | |
| have hm' : (g+1)*(3*n^3+128) ≤ 4*u^4 := by | |
| convert hm using 1; ring | |
| have hmajor : 133*u^4 ≤ u^5 := by | |
| calc | |
| _ ≤ u*u^4 := Nat.mul_le_mul_right _ (by omega) | |
| _ = _ := (pow_succ' _ _).symm | |
| unfold FixedSizeSampleCells.reserve | |
| nlinarith only [hm', hg3, h34, hu4, hu, hmajor] | |
| theorem four_cells {n g : ℕ} (I : Instance n) (hn : 10000 ≤ n) (hg : g ≤ n) | |
| (C : ℕ) (hC : 2^g ≤ C) (k m : Fin 4 → ℕ) (hk : ∀ i, k i ≤ g) | |
| (hm : ∀ i, m i ≤ 11*2^g) | |
| (s : (SeparatorFamilyExecution.coupling n g 4 k m hk).Seed) : | |
| (SeparatorSamplerCells.run n g 4 k m hk s).2 ≤ inputScale I^6*(C+1) := by | |
| have hu : n+1 ≤ inputScale I := by unfold inputScale; omega | |
| have hu0 : 10000 ≤ inputScale I := by omega | |
| have hr := reserve_le n g (inputScale I) hu hg hu0 | |
| have hp := Nat.mul_le_mul (Nat.mul_le_mul_left 11 hC) hr | |
| have hc : 44*inputScale I^5 ≤ inputScale I^6 := by | |
| calc | |
| _ ≤ inputScale I*inputScale I^5 := Nat.mul_le_mul_right _ (by omega) | |
| _ = _ := (pow_succ' _ _).symm | |
| have hmain := Nat.mul_le_mul_left C hc | |
| have hbase : 208 ≤ inputScale I^6 := by | |
| have hpow : inputScale I ≤ inputScale I^6 := by | |
| simpa only [pow_one] using pow_le_pow_right' (by omega : 1 ≤ inputScale I) | |
| (by decide : 1 ≤ 6) | |
| omega | |
| have h := SeparatorSamplerCells.four_cells n g k m hk hm s | |
| nlinarith only [h, hp, hmain, hbase] | |
| noncomputable def application {g : ℕ} (S : SeparatorExpectation.RoundedParameters g) | |
| (n : ℕ) (s : (S.samplerCoupling n).Seed) := | |
| SeparatorSamplerCells.run n g 4 S.e S.counts S.size_le s | |
| theorem application_value {g : ℕ} (S : SeparatorExpectation.RoundedParameters g) | |
| (n : ℕ) (s : (S.samplerCoupling n).Seed) : | |
| (application S n s).1 = SeparatorFamilyExecution.Application.run S n s := by | |
| exact SeparatorSamplerCells.run_value n g 4 S.e S.counts S.size_le s | |
| theorem application_cells {n g : ℕ} (I : Instance n) | |
| (S : SeparatorExpectation.RoundedParameters g) (hn : 10000 ≤ n) (hg : g ≤ n) | |
| (C : ℕ) (hC : 2^g ≤ C) (s : (S.samplerCoupling n).Seed) : | |
| (application S n s).2 ≤ inputScale I^6*(C+1) := | |
| four_cells I hn hg C hC S.e S.counts S.size_le S.counts_le s | |
| /-- The concrete top blocks obey a stronger table-capacity guard, hence | |
| all four sampled families and their unread seed tails fit the same cap. -/ | |
| theorem representation_cells {n : ℕ} (R : RepresentationSetup n) | |
| (h : R.PrimeHistory) (hn : 10000 ≤ n) | |
| (s : ((RepresentationSetup.separatorParameters h.1.1.2 (by omega)).samplerCoupling n).Seed) : | |
| (application (RepresentationSetup.separatorParameters h.1.1.2 (by omega)) n s).2 ≤ | |
| inputScale R.input^6*(compressionDictionaryCapacity n+1) := by | |
| have hg : 2*blockHalf n 0 ≤ n := by | |
| simp only [blockHalf, Matrix.cons_val_zero] | |
| omega | |
| have hpow : 2^(2*blockHalf n 0) ≤ compressionDictionaryCapacity n := by | |
| exact (Nat.pow_le_pow_right (by decide) | |
| (show 2*blockHalf n 0 ≤ 2*(2*blockHalf n 0) by omega)).trans | |
| (RepresentationSetup.separator_table_rate_le_capacity (n := n)) | |
| exact application_cells R.input _ hn hg _ hpow s | |
| end LowSpaceSubsetSum.SeparatorSamplerBounds | |
| end OAI | |