openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Representation /RepresentationEntrySoundness.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationParentStream | |
| namespace OAI | |
| /-! # Original indexed subset witnesses for retained parent entries -/ | |
| namespace LowSpaceSubsetSum | |
| namespace RepresentationUniverse | |
| variable {n : ℕ} | |
| def parentDomain (L : Fin 4 → RepresentationLeaf (Fin n)) : Finset (Fin n) := | |
| (topDomain L ∪ mixerDomain L) ∪ backgroundDomain L | |
| theorem entry_represents (a : Fin n → ℤ) (L : Fin 4 → RepresentationLeaf (Fin n)) | |
| (htl : Disjoint (topDomain L) (mixerDomain L)) | |
| (htb : Disjoint (topDomain L) (backgroundDomain L)) | |
| (hlb : Disjoint (mixerDomain L) (backgroundDomain L)) | |
| {x : ParentOccurrence (Fin n)} (hx : x ∈ parents L) (hv : Valid x) : | |
| Represents a (parentDomain L) (topDomain L) (entry a x) := by | |
| obtain ⟨ht, hl, hb⟩ := parent_part_subsets L hx | |
| have dtl := htl.mono ht hl | |
| have dtb := htb.mono ht hb | |
| have dlb := hlb.mono hl hb | |
| refine ⟨(topMask x ∪ localMask x) ∪ backgroundMask x, | |
| Finset.union_subset_union (Finset.union_subset_union ht hl) hb, ?_, ?_⟩ | |
| · rw [weight_union a (Finset.disjoint_union_left.mpr ⟨dtb, dlb⟩), weight_union a dtl] | |
| exact (valid_weight a hv).symm | |
| · ext i | |
| constructor | |
| · intro hi | |
| obtain ⟨hi, hiTop⟩ := Finset.mem_inter.mp hi | |
| rcases Finset.mem_union.mp hi with hi | hi | |
| · rcases Finset.mem_union.mp hi with hi | hi | |
| · exact hi | |
| · exact (Finset.disjoint_left.mp htl hiTop (hl hi)).elim | |
| · exact (Finset.disjoint_left.mp htb hiTop (hb hi)).elim | |
| · intro hi | |
| exact Finset.mem_inter.mpr | |
| ⟨Finset.mem_union_left _ (Finset.mem_union_left _ hi), ht hi⟩ | |
| end RepresentationUniverse | |
| namespace RepresentationParentData | |
| variable {n : ℕ} (D : RepresentationParentData n) | |
| theorem entry_represents (B : ℕ) | |
| (htl : Disjoint (RepresentationUniverse.topDomain D.leaves) | |
| (RepresentationUniverse.mixerDomain D.leaves)) | |
| (htb : Disjoint (RepresentationUniverse.topDomain D.leaves) | |
| (RepresentationUniverse.backgroundDomain D.leaves)) | |
| (hlb : Disjoint (RepresentationUniverse.mixerDomain D.leaves) | |
| (RepresentationUniverse.backgroundDomain D.leaves)) | |
| (e : Entry (Fin n)) (he : e ∈ (D.entries B).toList) : | |
| Represents D.weights (RepresentationUniverse.parentDomain D.leaves) | |
| (RepresentationUniverse.topDomain D.leaves) e := by | |
| obtain ⟨x, hx, hv, rfl, _⟩ := D.entries_provenance B e he | |
| exact RepresentationUniverse.entry_represents D.weights D.leaves htl htb hlb hx hv | |
| theorem entry_bounded (B : ℕ) | |
| (htl : Disjoint (RepresentationUniverse.topDomain D.leaves) | |
| (RepresentationUniverse.mixerDomain D.leaves)) | |
| (htb : Disjoint (RepresentationUniverse.topDomain D.leaves) | |
| (RepresentationUniverse.backgroundDomain D.leaves)) | |
| (hlb : Disjoint (RepresentationUniverse.mixerDomain D.leaves) | |
| (RepresentationUniverse.backgroundDomain D.leaves)) | |
| (e : Entry (Fin n)) (he : e ∈ (D.entries B).toList) : | |
| WeightEncoding.BoundedWeight D.input e.value := by | |
| obtain ⟨S, _, hw, _⟩ := D.entry_represents B htl htb hlb e he | |
| rw [← hw] | |
| exact WeightEncoding.signed_weight_bounded D.input D.signs S | |
| end RepresentationParentData | |
| end LowSpaceSubsetSum | |
| end OAI | |