| import Mathlib |
| import OAI.Computability.MinUncut.Search.FiniteEnumeration |
| import OAI.Computability.MinUncut.Estimates.WeightedTestLaw |
| import OAI.Computability.MinUncut.Encoding.QuestionEncoding |
|
|
| namespace OAI |
|
|
| section |
| namespace MinUncut.Outer |
| open MinUncut.Inner MinUncut.FiniteGaussian MinUncut.FiniteProof OuterSmoothness |
| open MinUncutGames.Foundations.Hastad.SourceOccurrences |
|
|
| namespace SampleEnumeration |
|
|
| def fin (n : ℕ) : Enumeration (Fin n) := Enumeration.ofEncoding (Encoding.fin n) |
| def field : Enumeration F₂ := Enumeration.ofEncoding fieldEncoding |
|
|
| def point (m n : ℕ) : Enumeration (Point m n) := (fin m).function (fin n) |
| def row (m n : ℕ) : Enumeration (Row m n) := |
| (fin m).sigma (fun i => ((fin m).subtype (fun j => j≠i)).function (fin n)) |
|
|
| local instance codeEq (m n : ℕ) : DecidableEq (Code m n) := |
| Subtype.instDecidableEq |
|
|
| def code (m n : ℕ) : Enumeration (Code m n) := |
| ((row m n).function field).image (faceSum m n).rangeRestrict (by |
| intro z |
| obtain ⟨f,hf⟩ := z.property |
| exact ⟨f,Subtype.ext hf⟩) |
|
|
| local instance codeType (m n : ℕ) : Fintype (Code m n) := |
| Fintype.ofList (code m n).values (code m n).complete |
|
|
| abbrev Slots {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) := |
| Σ j, Fin (secondSlotCount (Q j)) |
|
|
| def slots {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) : Enumeration (Slots Q) := |
| (fin t).sigma (fun j => fin (secondSlotCount (Q j))) |
|
|
| def formsPiEquiv (A : Type) [Fintype A] [DecidableEq A] : |
| Forms (A → F₂) ≃ F₂ × (A → F₂) := |
| (AffineMap.toConstProdLinearMap F₂).toEquiv.trans |
| (Equiv.prodCongr (Equiv.refl F₂) (LinearEquiv.piRing F₂ F₂ A F₂).toEquiv) |
|
|
| def formsCurryEquiv (A : Type) (B : A → Type) : |
| Forms (∀a, B a → F₂) ≃ Forms ((Σa, B a) → F₂) := |
| AffineEquiv.arrowCongrEquiv |
| (LinearEquiv.piCurry F₂ (fun (_ : A) (_ : B _) => F₂)).symm.toAffineEquiv |
| (AffineEquiv.refl F₂ F₂) |
|
|
| def formsEquiv {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) : |
| Forms (SecondAlphabet Q) ≃ F₂ × (Slots Q → F₂) := |
| (formsCurryEquiv (Fin t) (fun j => Fin (secondSlotCount (Q j)))).trans (formsPiEquiv (Slots Q)) |
|
|
| local instance formEq {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) : |
| DecidableEq (Forms (SecondAlphabet Q)) := |
| fun f g => decidable_of_iff (formsEquiv Q f=formsEquiv Q g) (formsEquiv Q).injective.eq_iff |
|
|
| def forms {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) : Enumeration (Forms (SecondAlphabet Q)) := |
| (field.pair ((slots Q).function field)).image (formsEquiv Q).symm (formsEquiv Q).symm.surjective |
|
|
| def arrays {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) (m n : ℕ) : |
| Enumeration (FaceArray (SecondAlphabet Q) m n) := (row m n).function (forms Q) |
|
|
| def coordinates (m n : ℕ) : Enumeration (TestCoordinates m n) := |
| Enumeration.ofEncoding ((point m n).encoding.sum ((point m n).encoding.sum (code m n).encoding)) |
|
|
| def localSample {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) (m n : ℕ) (g : GridData) : |
| Enumeration (LocalSample (SecondAlphabet Q) m n g) := by |
| letI : DecidableEq (Forms (SecondAlphabet Q)) := formEq Q |
| letI : DecidableEq (FaceArray (SecondAlphabet Q) m n) := inferInstanceAs (DecidableEq (Row m n → Forms (SecondAlphabet Q))) |
| exact (arrays Q m n).pair ((arrays Q m n).pair ((code m n).pair ((coordinates m n).function (fin g.L)))) |
|
|
| def finsets (t : ℕ) : Enumeration (Finset (Fin t)) := |
| ((fin t).function (Enumeration.ofEncoding Encoding.bool)).image |
| (fun f => Finset.univ.filter (fun j => f j=true)) (by |
| intro S |
| refine ⟨fun j => decide (j∈S),?_⟩ |
| ext j |
| simp) |
|
|
| def fixedSets (t k : ℕ) : Enumeration (FixedSets (Fin t) k) := |
| (finsets t).subtype (fun H => H.card=k) |
|
|
| def outerSample (t s k : ℕ) : Enumeration (OuterSample (Fin t) (Fin s) k) := |
| (fixedSets t k).pair (((fin t).function (fin s)).pair ((fin t).function (fin 3))) |
|
|
| local instance localEq {N t : ℕ} (Q : Fin t → SecondQuestion (Fin N)) (m n : ℕ) (g : GridData) : |
| DecidableEq (LocalSample (SecondAlphabet Q) m n g) := by |
| letI : DecidableEq (Forms (SecondAlphabet Q)) := formEq Q |
| letI : DecidableEq (FaceArray (SecondAlphabet Q) m n) := |
| inferInstanceAs (DecidableEq (Row m n → Forms (SecondAlphabet Q))) |
| exact fun f h => decidable_of_iff |
| (f.1=h.1 ∧ f.2.1=h.2.1 ∧ f.2.2.1=h.2.2.1 ∧ f.2.2.2=h.2.2.2) |
| (by constructor <;> intro he <;> aesop) |
|
|
| def second {N s t : ℕ} (eqs : Fin s → Equation (Fin N)) {k : ℕ} |
| (o : OuterSample (Fin t) (Fin s) k) : Fin t → SecondQuestion (Fin N) := |
| fun j => if j∈o.1.val then Sum.inl ((eqs (o.2.1 j)).names (o.2.2 j)) else Sum.inr (eqs (o.2.1 j)) |
|
|
| lemma second_eq {N s t : ℕ} (eqs : Fin s → Equation (Fin N)) {k : ℕ} |
| (o : OuterSample (Fin t) (Fin s) k) : second eqs o=sampleSecond eqs o := by |
| funext j |
| simp [second,sampleSecond,secondQuestion,sampleFirst,sampleHidden,hiddenSet,slotQuestion] |
|
|
| def semanticLocal {N s t : ℕ} (eqs : Fin s → Equation (Fin N)) {k : ℕ} |
| (o : OuterSample (Fin t) (Fin s) k) (m n : ℕ) (g : GridData) : |
| Enumeration (LocalSample (SecondAlphabet (sampleSecond eqs o)) m n g) := |
| (second_eq eqs o) ▸ localSample (second eqs o) m n g |
|
|
| local instance semanticLocalEq {N s t : ℕ} (eqs : Fin s → Equation (Fin N)) {k : ℕ} |
| (o : OuterSample (Fin t) (Fin s) k) (m n : ℕ) (g : GridData) : |
| DecidableEq (LocalSample (SecondAlphabet (sampleSecond eqs o)) m n g) := |
| (second_eq eqs o) ▸ localEq (second eqs o) m n g |
|
|
| def elementary {N s t : ℕ} (eqs : Fin s → Equation (Fin N)) (k m n : ℕ) (g : GridData) : |
| Enumeration (Elementary (I := Fin t) eqs k m n g) := |
| (outerSample t s k).sigma (fun o => semanticLocal eqs o m n g) |
|
|
| def tests : Enumeration Test := |
| Enumeration.ofList [.first,.second,.third,.fourth] (by intro j; cases j <;> simp) |
|
|
| def all {N s t : ℕ} (eqs : Fin s → Equation (Fin N)) (k m n : ℕ) (g : GridData) : |
| Enumeration (Test × Elementary (I := Fin t) eqs k m n g) := |
| tests.pair (elementary eqs k m n g) |
|
|
| end SampleEnumeration |
| end MinUncut.Outer |
|
|
| end |
|
|
| end OAI |
|
|
