| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Tapes.FixedAlphabetCompiler |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.FixedEncoding |
| variable (Γ : Type*) [Fintype Γ] [Inhabited Γ] |
| noncomputable def enc (a : Γ) : List.Vector Bool (Fintype.card Γ) := by |
| classical |
| exact List.Vector.ofFn (fun i=>decide (a≠default ∧ Fintype.equivFin Γ a=i)) |
| lemma blank : enc Γ default=List.Vector.replicate (Fintype.card Γ) false := by |
| classical |
| apply List.Vector.ext |
| intro i |
| simp [enc] |
| lemma injective : Function.Injective (enc Γ) := by |
| classical |
| intro a b hab |
| by_cases ha : a=default |
| · by_cases hb : b=default |
| · exact ha.trans hb.symm |
| · have hh:=congrArg (fun v : List.Vector Bool (Fintype.card Γ)=>v.get (Fintype.equivFin Γ b)) hab |
| simp [enc,ha,hb] at hh |
| · have hh:=congrArg (fun v : List.Vector Bool (Fintype.card Γ)=>v.get (Fintype.equivFin Γ a)) hab |
| simp only [enc,List.Vector.get_ofFn] at hh |
| have ht : decide (b≠default ∧ Fintype.equivFin Γ b=Fintype.equivFin Γ a)=true := by |
| simpa [ha] using hh.symm |
| exact ((Fintype.equivFin Γ).injective (of_decide_eq_true ht).2).symm |
| noncomputable def dec : List.Vector Bool (Fintype.card Γ) → Γ := Function.invFun (enc Γ) |
| lemma left_inverse (a : Γ) : dec Γ (enc Γ a)=a := |
| Function.leftInverse_invFun (injective Γ) a |
| lemma bitsWord_length {Γ : Type*} {w : ℕ} (e : Γ → List.Vector Bool w) (input : List Γ) : |
| (FixedAlphabetCompiler.bitsWord e input).length=w*input.length := by |
| induction input with |
| | nil => simp [FixedAlphabetCompiler.bitsWord] |
| | cons a input ih => |
| simp only [FixedAlphabetCompiler.bitsWord,List.flatMap_cons,List.length_append] at ih ⊢ |
| rw [show (e a).toList.length=w from (e a).2,ih,List.length_cons,Nat.mul_add,Nat.mul_one] |
| omega |
| end MatchingFPRAS.FixedEncoding |
|
|
| end OAI |
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|
