openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Representation /RepresentationLeafArrays.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.StoredLeafPreparation | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationPrefixSource | |
| namespace OAI | |
| /-! # Preparing every leaf from the accepted stored prefix -/ | |
| namespace LowSpaceSubsetSum.RepresentationLeafArrays | |
| variable {n : ℕ} | |
| abbrev Leaves (n : ℕ) := Fin 2 → Fin 4 → RepresentationLeaf (Fin n) | |
| def run (I : Instance n) (data : RepresentationPrefixArrays.Data n) : Option (Leaves n) × ℕ := | |
| let ordered := StoredOrderedMixers.run data.layout | |
| match ordered.1 with | |
| | none => (none, ordered.2 + 1) | |
| | some domains => | |
| let background := RepresentationBackgroundPreparation.run I data.layout.permutation data.guesses | |
| match background.1 with | |
| | none => (none, ordered.2 + background.2 + 2) | |
| | some rows => | |
| let out := StoredLeafPreparation.run domains data.partitions rows data.guesses | |
| (some out.1, ordered.2 + background.2 + out.2 + 3) | |
| theorem run_of_stages (I : Instance n) (data : RepresentationPrefixArrays.Data n) | |
| (domains : StoredOrderedMixers.Domains n) (background : Vector (List (Fin n)) 8) | |
| (hd : (StoredOrderedMixers.run data.layout).1 = some domains) | |
| (hb : (RepresentationBackgroundPreparation.run I data.layout.permutation data.guesses).1 = some background) : | |
| (run I data).1 = some (StoredLeafPreparation.run domains data.partitions background data.guesses).1 := by | |
| simp only [run, hd, hb] | |
| theorem work (I : Instance n) (data : RepresentationPrefixArrays.Data n) | |
| (hl : ∀ j : Fin 3, data.layout.domains[j.val].length ≤ n) : | |
| (run I data).2 ≤ 3000 * (n + 1) ^ 2 := by | |
| have ho := StoredOrderedMixers.work data.layout hl | |
| have hb := RepresentationBackgroundPreparation.work I data.layout.permutation data.guesses | |
| simp only [run] | |
| split | |
| · dsimp only | |
| nlinarith only [ho, Nat.zero_le (n ^ 2)] | |
| next domains hd => | |
| split | |
| · dsimp only | |
| nlinarith only [ho, hb, Nat.zero_le (n ^ 2)] | |
| next rows hr => | |
| have hw := StoredLeafPreparation.work domains data.partitions rows data.guesses | |
| (RepresentationBackgroundPreparation.lengths I data.layout.permutation data.guesses rows hr) | |
| change _ + _ + _ + 3 ≤ _ | |
| nlinarith only [ho, hb, hw, Nat.zero_le (n ^ 2)] | |
| noncomputable section | |
| theorem complete (I : Instance n) | |
| (s : (canonicalRepresentationData I).zeroSetup.partitionHistoryCoupling.Seed) | |
| (data : RepresentationPrefixArrays.Data n) | |
| (h : (RepresentationPrefixArrays.run (canonicalRepresentationData I) s).1 = some data) | |
| (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000) : | |
| ∃ history leaves, | |
| (canonicalRepresentationData I).zeroSetup.partitionHistoryCoupling.bounded s = some history ∧ | |
| (run I data).1 = some leaves ∧ | |
| ∀ (side : Fin 2) (i : Fin 4), | |
| leaves side i = (canonicalRepresentationData I).zeroSetup.rawLeaf | |
| history.1.1 history.1.2 history.2 | |
| ((canonicalRepresentationData I).zeroSetup.allocatedBackground history.1.1 history.1.2 hn hlog) | |
| side i := by | |
| obtain ⟨history, hh, _, hv, hg, hl, _⟩ := | |
| RepresentationPrefixArrays.source (canonicalRepresentationData I) s data h | |
| obtain ⟨hL, _, _⟩ := RepresentationPrefixArrays.components (canonicalRepresentationData I) s data h | |
| obtain ⟨domains, hd, hds⟩ := StoredOrderedMixers.complete (canonicalRepresentationData I) | |
| s.1.1 data.layout hL history.1.1.1 hv history.1.1.2 | |
| obtain ⟨background, hb, hbs⟩ := RepresentationBackgroundPreparation.complete I | |
| data.layout.permutation history.1.1.1 hv history.1.1.2 data.guesses history.1.2 hg hn hlog | |
| refine ⟨history, (StoredLeafPreparation.run domains data.partitions background data.guesses).1, | |
| hh, run_of_stages I data domains background hd hb, ?_⟩ | |
| intro side i | |
| exact StoredLeafPreparation.value (canonicalRepresentationData I).zeroSetup history.1.1 | |
| history.1.2 history.2 _ domains data.partitions background data.guesses hds hl hbs hg side i | |
| end | |
| end LowSpaceSubsetSum.RepresentationLeafArrays | |
| end OAI | |