openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Compression /CompressionCallbackBounds.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionCallbackCells | |
| namespace OAI | |
| /-! # Capacity-scaled bounds for the actual compression callbacks -/ | |
| namespace LowSpaceSubsetSum.CompressionCallbackBounds | |
| variable {n : ℕ} | |
| private theorem polynomial (I : Instance n) (hn : 10000 ≤ n) : | |
| 1000*(inputScale I+1) ≤ inputScale I^4 := by | |
| have hu : 10000 ≤ inputScale I := by unfold inputScale; omega | |
| have h2 : 1000*(inputScale I+1) ≤ inputScale I^2 := by nlinarith only [hu] | |
| exact h2.trans (Nat.pow_le_pow_right (by omega) (by decide : 2 ≤ 4)) | |
| theorem dictionary (I : Instance n) (hn : 10000 ≤ n) (C : ℕ) | |
| (r : CompressionCallbacks.Dictionary I C) : | |
| CompressionCallbackCells.dictionaryCells r ≤ inputScale I^4*(C+1) := by | |
| have h := CompressionCallbackCells.dictionaryCells_bound I C r | |
| have hp := Nat.mul_le_mul_right (C+1) (polynomial I hn) | |
| nlinarith only [h,hp] | |
| theorem report (I : Instance n) (hn : 10000 ≤ n) (C : ℕ) (x y : ℤ) | |
| (r : CompressionCallbacks.Dictionary I C) : | |
| (CompressionCallbackCells.report I C x y r).2 ≤ inputScale I^4*(C+1) := by | |
| have h := CompressionCallbackCells.report_cells I C x y r | |
| have hp := Nat.mul_le_mul_right (C+1) (polynomial I hn) | |
| nlinarith only [h,hp] | |
| theorem probe (I : Instance n) (hn : 10000 ≤ n) (C : ℕ) | |
| (tree : BinaryTrie.Tree) (t x y : ℤ) (found : Bool) : | |
| (CompressionCallbackCells.probe I tree t x y found).2 ≤ inputScale I^4*(C+1) := by | |
| have h := CompressionCallbackCells.probe_cells I tree t x y found | |
| have hp := Nat.mul_le_mul_right (C+1) (polynomial I hn) | |
| nlinarith only [h,hp] | |
| end LowSpaceSubsetSum.CompressionCallbackBounds | |
| end OAI | |