File size: 1,840 Bytes
e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 | 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
|