openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Transport /BooleanEnumeration.lean
| import OAI.Combinatorics.ContingencyTables.Machines.CompletionEnumerationProgram | |
| namespace OAI | |
| /-! # Enumerating only the short proposal word -/ | |
| namespace ContingencyTables.DensePrograms | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped | |
| open scoped BigOperators Classical | |
| def digitBool (a : ℕ) : Bool := decide (a=1) | |
| def digitBoolRealizer : Realizer digitBool := composition (pair identity (constant 1)) natEqual | |
| lemma polynomial_digitBool : PolynomialTime digitBoolRealizer := | |
| polynomial_composition (polynomial_pair polynomial_identity (polynomial_constant 1)) | |
| polynomial_natEqual | |
| def booleanWords (xs : List Bool) : List (List Bool) := | |
| (gridWords (xs,2)).map (List.map digitBool) | |
| def booleanWordsRealizer : Realizer booleanWords := | |
| composition (composition (pair identity (constant 2)) gridWordsRealizer) | |
| (map (map digitBoolRealizer)) | |
| lemma work_booleanWords : WorkBound booleanWordsRealizer | |
| (fun xs => gridStateCount (xs,2)+weight xs) := by | |
| have hi : WorkBound (pair (identity (A := List Bool)) (constant 2)) | |
| (fun xs => gridStateCount (xs,2)+weight xs) := | |
| work_of_polynomial (polynomial_pair polynomial_identity (polynomial_constant 2)) | |
| Polynomial.X (fun xs => by simp) | |
| have hg : WorkBound (composition (pair (identity (A := List Bool)) (constant 2)) | |
| gridWordsRealizer) (fun xs => gridStateCount (xs,2)+weight xs) := by | |
| apply work_composition hi work_gridWords (Polynomial.X+10) | |
| intro xs | |
| simp only [Polynomial.eval_add,Polynomial.eval_X,Polynomial.eval_ofNat,weight_prod,id_eq] | |
| have hc : weight (2:ℕ)≤9 := by simpa using weight_nat_numeric 2 | |
| omega | |
| unfold booleanWordsRealizer | |
| exact work_then_polynomial hg (polynomial_map (polynomial_map polynomial_digitBool)) | |
| def booleanEnumeration (s : ℕ) : Fin (2^s) ≃ (Fin s → Bool) := | |
| finFunctionFinEquiv.symm.trans (Equiv.piCongrRight (fun _ => finTwoEquiv)) | |
| lemma booleanWords_of_length (xs : List Bool) (s : ℕ) (hs : xs.length=s) : | |
| booleanWords xs=List.ofFn (fun k => List.ofFn (booleanEnumeration s k)) := by | |
| unfold booleanWords | |
| rw [gridWords_finFunctions xs 1 s hs,List.map_ofFn] | |
| apply congrArg List.ofFn | |
| funext k | |
| dsimp only [Function.comp_apply] | |
| rw [List.map_ofFn] | |
| apply congrArg List.ofFn | |
| funext i | |
| change digitBool ((finFunctionFinEquiv.symm k) i).val= | |
| finTwoEquiv ((finFunctionFinEquiv.symm k) i) | |
| have hb (a : Fin 2) : digitBool a.val=finTwoEquiv a := by | |
| fin_cases a <;> rfl | |
| exact hb _ | |
| lemma sum_booleanWords (xs : List Bool) (s : ℕ) (hs : xs.length=s) | |
| (w : List Bool → ℕ) : | |
| ((booleanWords xs).map w).sum=∑ bits : Fin s → Bool,w (List.ofFn bits) := by | |
| rw [booleanWords_of_length xs s hs,List.map_ofFn,List.sum_ofFn] | |
| exact (booleanEnumeration s).sum_comp (fun bits => w (List.ofFn bits)) | |
| end ContingencyTables.DensePrograms | |
| end OAI | |