openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Dense /DenseBooleanWalkProgram.lean
| import OAI.Combinatorics.ContingencyTables.Machines.BooleanChunkProgram | |
| import OAI.Combinatorics.ContingencyTables.Dense.DenseWalkSemantics | |
| namespace OAI | |
| /-! # A polynomial program for the complete flat-word dense walk -/ | |
| namespace ContingencyTables.DensePrograms | |
| open MatchingFPRAS.TreeTyped GridBoundary ProfilePrograms | |
| def denseWidthRealizer : Realizer denseStepBits := by | |
| let index := composition (composition (pair (constant 8) identity) multiply) ceilLogTwoRealizer | |
| let coins := composition (pair (constant 8) fifthPower) multiply | |
| exact composition (pair index coins) add | |
| theorem polynomial_denseWidth : PolynomialTime denseWidthRealizer := by | |
| unfold denseWidthRealizer | |
| exact polynomial_composition (polynomial_pair | |
| (polynomial_composition (polynomial_composition | |
| (polynomial_pair (polynomial_constant 8) polynomial_identity) polynomial_multiply) | |
| polynomial_ceilLogTwo) | |
| (polynomial_composition (polynomial_pair (polynomial_constant 8) polynomial_fifthPower) | |
| polynomial_multiply)) polynomial_add | |
| def flatWordWalk (z : FlatExponentData×List Bool) : FlatExponentData := | |
| walkWords (z.1,wordChunks (denseStepBits z.1.2,z.2)) | |
| def flatWordWalkRealizer : Realizer flatWordWalk := | |
| composition (pair first (composition (pair | |
| (composition (composition first second) denseWidthRealizer) second) wordChunksRealizer)) | |
| walkWordsRealizer | |
| theorem polynomial_flatWordWalk : PolynomialTime flatWordWalkRealizer := | |
| polynomial_composition (polynomial_pair polynomial_first (polynomial_composition (polynomial_pair | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_second) polynomial_denseWidth) | |
| polynomial_second) polynomial_wordChunks)) polynomial_walkWords | |
| lemma denseStepBits_pos {d : ℕ} (hd : 0<d) : 0<denseStepBits d := by | |
| have h := Nat.mul_pos (by decide : 0<8) (pow_pos hd 5) | |
| exact h.trans_le (Nat.le_add_left _ _) | |
| noncomputable section | |
| open scoped Classical | |
| /-- The literal flat-bit program computes exactly the finite Boolean walk. | |
| The polynomial runtime theorem above applies to every input word. -/ | |
| theorem flatWordWalk_code {m n d : ℕ} (hd : 0<d) | |
| (R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ) | |
| (a : Grid (I := Fin (m*n)) (4*d^8)) (steps : ℕ) | |
| (bits : Fin (steps*denseStepBits d) → Bool) : | |
| flatWordWalk ((((marginCode R,marginCode P),gridNatCode a),d),List.ofFn bits)= | |
| (((marginCode R,marginCode P), | |
| gridNatCode (denseBooleanWalk R P d finProdFinEquiv a steps bits)),d) := by | |
| dsimp only [flatWordWalk] | |
| rw [wordChunks_ofFn (denseStepBits_pos hd) bits] | |
| exact walkWords_ofFn R P a steps (gridWordEquiv steps (denseStepBits d) bits) | |
| end | |
| end ContingencyTables.DensePrograms | |
| end OAI | |