openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Local /LocalBadCoverSpecifications.lean
| import OAI.Combinatorics.BalancedRyser.Local.LocalSpecificationEncoding | |
| import OAI.Combinatorics.BalancedRyser.Probability.FiniteDependencyRank | |
| namespace OAI | |
| namespace Balanced.Local.ConditionSpec | |
| noncomputable section | |
| open scoped Classical | |
| variable {b κ n : ℕ} | |
| theorem exists_goodSpecification_of_acyclic_fixed | |
| (u : Position b → Position b) (I T : Finset (Position b)) | |
| (hTI : T ⊆ I) (hcard : T.card = n) (hfix : ∀ v ∈ T, u v = v) | |
| (code : Position b → Code b κ) | |
| (hwf : WellFounded (fun x y : {v // v ∈ T} => | |
| y.val ∈ underlyingAnchors u (code x.val))) | |
| (E : Position b → Code b κ → Prop) (hE : ∀ v ∈ T, E v (code v)) : | |
| ∃ σ : Fin n → Position b × Code b κ, | |
| GoodSpecification u I σ ∧ | |
| (∀ k, E (σ k).1 (σ k).2) ∧ | |
| Finset.univ.image (fun k => (σ k).1) = T := by | |
| obtain ⟨rank, hrank⟩ := exists_nat_rank_on_finset T | |
| (fun v w => w ∈ underlyingAnchors u (code v)) hwf | |
| apply exists_goodSpecification_of_finset u I T hTI hcard | |
| (fun v hv w hw h => by simpa only [hfix v hv, hfix w hw] using h) | |
| code rank _ E hE | |
| intro v hv w hw hdep | |
| rw [hfix v hv, hfix w hw] | |
| exact hrank v hv w hw (by simpa only [underlyingAnchors, hfix w hw] using hdep) | |
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] | |
| theorem badCover_zero_reuse_exists_goodSpecification | |
| (P : EqualityPattern b 0) (a : DirectionAssignment F b 0) | |
| (bits : Bits b) (p : Point F) (M : MergedPairs F) | |
| (hb : 3 ≤ b) (ha : Eligible P a) | |
| (hadm : Admissible P a bits p M) (hbad : BadCover P a bits p M) : | |
| ∃ σ : Fin 3 → Position b × Code b 0, | |
| GoodSpecification id Finset.univ σ ∧ | |
| ∀ k, Holds P a bits p M (σ k).1 (σ k).2 := by | |
| obtain ⟨T, code, hcard, hT, hwf⟩ := | |
| badCover_zero_reuse_exists_acyclic_spec P a bits p M hb ha hadm hbad | |
| obtain ⟨σ, hσ, hE, _⟩ := exists_goodSpecification_of_acyclic_fixed | |
| id Finset.univ T (Finset.subset_univ _) hcard (fun _ _ => rfl) code | |
| (by simpa only [underlyingAnchors, Finset.image_id] using hwf) | |
| (Holds P a bits p M) (fun v hv => (hT v hv).1) | |
| exact ⟨σ, hσ, hE⟩ | |
| theorem badCover_one_reuse_exists_goodSpecification | |
| (P : EqualityPattern b 1) (a : DirectionAssignment F b 1) | |
| (bits : Bits b) (p : Point F) (M : MergedPairs F) | |
| (hb : 3 ≤ b) (ha : Eligible P a) | |
| (hadm : Admissible P a bits p M) (hbad : BadCover P a bits p M) | |
| (root child : Fin b) (r s : Fin 4) | |
| (hshared : P.classOf root r = P.classOf child s) : | |
| let u := ParentMap.underlying root child r s (bits root r) (bits child s) | |
| ∃ σ : Fin 2 → Position b × Code b 1, | |
| GoodSpecification u (Finset.univ \ ParentMap.dependent child s) σ ∧ | |
| ∀ k, Holds P a bits p M (σ k).1 (σ k).2 := by | |
| dsimp only | |
| let u := ParentMap.underlying root child r s (bits root r) (bits child s) | |
| obtain ⟨T, code, hcard, hT, hwf⟩ := | |
| badCover_one_reuse_exists_acyclic_spec P a bits p M hb ha hadm hbad | |
| root child r s hshared | |
| have hTI : T ⊆ Finset.univ \ ParentMap.dependent child s := by | |
| intro v hv | |
| exact Finset.mem_sdiff.mpr ⟨Finset.mem_univ _, (hT v hv).1⟩ | |
| have hfix : ∀ v ∈ T, u v = v := by | |
| intro v hv | |
| exact ParentMap.underlying_fixed root child r s (bits root r) (bits child s) | |
| (hT v hv).1 | |
| obtain ⟨σ, hσ, hE, _⟩ := exists_goodSpecification_of_acyclic_fixed u | |
| (Finset.univ \ ParentMap.dependent child s) T hTI hcard hfix code hwf | |
| (Holds P a bits p M) (fun v hv => (hT v hv).2.1) | |
| exact ⟨σ, hσ, hE⟩ | |
| end | |
| end Balanced.Local.ConditionSpec | |
| end OAI | |