openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Dense /DenseEnumeratedMatrix.lean
| import OAI.Combinatorics.ContingencyTables.Dense.DenseMatrixWalk | |
| import OAI.Combinatorics.ContingencyTables.Transport.FiniteGridWords | |
| namespace OAI | |
| /-! # Computing every state of the dense transition table -/ | |
| namespace ContingencyTables.DensePrograms | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped ProfilePrograms GridBoundary | |
| open scoped BigOperators | |
| abbrev EnumeratedMatrixInput := CountMatrixContext×List Bool | |
| def enumerationData (z : EnumeratedMatrixInput) : GridEnumerationData := | |
| (z.2,4*z.1.2.2^8) | |
| def enumerationDataRealizer : Realizer enumerationData := by | |
| let d : Realizer (fun z : EnumeratedMatrixInput => z.1.2.2) := | |
| composition (composition first second) second | |
| let K := composition (pair (constant 4) (composition d eighthPower)) multiply | |
| exact pair second K | |
| lemma polynomial_enumerationData : PolynomialTime enumerationDataRealizer := by | |
| have hd : PolynomialTime (composition (composition | |
| (first (A := CountMatrixContext) (B := List Bool)) second) second) := | |
| polynomial_composition (polynomial_composition polynomial_first polynomial_second) | |
| polynomial_second | |
| unfold enumerationDataRealizer | |
| exact polynomial_pair polynomial_second (polynomial_composition | |
| (polynomial_pair (polynomial_constant 4) (polynomial_composition hd polynomial_eighthPower)) | |
| polynomial_multiply) | |
| def enumeratedMatrixInput (z : EnumeratedMatrixInput) : CountMatrixData := | |
| (z.1,gridWords (enumerationData z)) | |
| def enumeratedMatrixInputRealizer : Realizer enumeratedMatrixInput := | |
| pair first (composition enumerationDataRealizer gridWordsRealizer) | |
| def enumerationMeasure (z : EnumeratedMatrixInput) : ℕ := | |
| gridStateCount (enumerationData z)+weight z | |
| lemma work_enumeratedMatrixInput : WorkBound enumeratedMatrixInputRealizer enumerationMeasure := by | |
| have hp : WorkBound enumerationDataRealizer enumerationMeasure := | |
| work_of_polynomial polynomial_enumerationData Polynomial.X (fun z => by | |
| simp only [enumerationMeasure,Polynomial.eval_X]; omega) | |
| obtain ⟨p,hp'⟩ := hp | |
| have hg : WorkBound (composition enumerationDataRealizer gridWordsRealizer) | |
| enumerationMeasure := by | |
| apply work_composition ⟨p,hp'⟩ work_gridWords (Polynomial.X+p) | |
| intro z | |
| have hw := hp' z | |
| simp only [enumerationMeasure,Polynomial.eval_add,Polynomial.eval_X] | |
| change gridStateCount (enumerationData z)+weight (enumerationData z) ≤ _ | |
| change _≤p.eval (enumerationMeasure z) at hw | |
| unfold enumerationMeasure at hw | |
| omega | |
| unfold enumeratedMatrixInputRealizer | |
| exact work_pair (work_of_polynomial polynomial_first Polynomial.X (fun z => by | |
| simp only [enumerationMeasure,Polynomial.eval_X]; omega)) hg Polynomial.X | |
| (fun z => by simp only [enumerationMeasure,Polynomial.eval_X]; omega) | |
| def enumeratedMatrix (z : EnumeratedMatrixInput) : List (List ℕ) := | |
| countMatrix (enumeratedMatrixInput z) | |
| def enumeratedMatrixRealizer : Realizer enumeratedMatrix := | |
| composition enumeratedMatrixInputRealizer countMatrixRealizer | |
| /-- All states are generated from the dimension marker and grid size. | |
| The work is polynomial in their count and the supplied binary data. -/ | |
| theorem work_enumeratedMatrix : WorkBound enumeratedMatrixRealizer enumerationMeasure := | |
| work_then_polynomial work_enumeratedMatrixInput polynomial_countMatrix | |
| theorem enumeratedMatrix_actual {m n d : ℕ} (R : Option (Fin m) → ℕ) | |
| (P : Option (Fin n) → ℕ) (hq : m*n≤d) | |
| (xs dims : List Bool) (hxs : xs.length=8*d^5) (hdims : dims.length=m*n) : | |
| enumeratedMatrix ((((xs,2^denseStepBits d),((marginCode R,marginCode P),d)),dims))= | |
| List.ofFn (fun j => List.ofFn (fun i => | |
| denseTransitionCount R P d finProdFinEquiv | |
| (gridEnumeration (m*n) (4*d^8) i) (gridEnumeration (m*n) (4*d^8) j))) := by | |
| unfold enumeratedMatrix enumeratedMatrixInput enumerationData | |
| rw [gridWords_eq_length dims (4*d^8) (m*n) hdims] | |
| exact countMatrix_actual R P hq (gridEnumeration (m*n) (4*d^8)) xs hxs | |
| end ContingencyTables.DensePrograms | |
| end OAI | |