| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Machines.TreeSum |
| import OAI.Combinatorics.MatchingCount.Machines.TreeClosureReindex |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.TreeTyped.Numbered |
| variable {A : Type} [Fintype A] [LinearOrder A] |
| def index (a : A) : ℕ := ((UnitRefresh.orderedEnum A).symm a).val |
| instance (priority := 100) codec : Codec A := ⟨fun a=>encode (index a)⟩ |
| lemma index_injective : Function.Injective (index (A:=A)) := by |
| intro a b h |
| exact (UnitRefresh.orderedEnum A).symm.injective (Fin.ext h) |
| lemma index_le (a b : A) : index a ≤ index b ↔ a ≤ b := by |
| change (UnitRefresh.orderedEnum A).symm a ≤ (UnitRefresh.orderedEnum A).symm b ↔ _ |
| rw [←not_lt,←not_lt,←UnitRefresh.orderedEnum_lt] |
| simp |
|
|
| def equal : Realizer (fun x : A×A=>decide (x.1=x.2)) := |
| natEqual.recode (fun x : A×A=>(index x.1,index x.2)) _ (fun _=>rfl) (fun x=>by |
| simp only [(index_injective (A:=A)).eq_iff]) |
| def lessEqual : Realizer (fun x : A×A=>decide (x.1≤x.2)) := |
| leNat.recode (fun x : A×A=>(index x.1,index x.2)) _ (fun _=>rfl) (fun x=>by |
| simp only [index_le]) |
| lemma polynomial_equal : PolynomialTime (equal (A:=A)) := |
| polynomial_recode _ _ _ _ polynomial_natEqual |
| lemma polynomial_lessEqual : PolynomialTime (lessEqual (A:=A)) := |
| polynomial_recode _ _ _ _ polynomial_leNat |
| @[simp] lemma equal_program : (equal (A:=A)).program=natEqual.program := rfl |
| @[simp] lemma lessEqual_program : (lessEqual (A:=A)).program=leNat.program := rfl |
|
|
| lemma index_orderedEnum (i : Fin (Fintype.card A)) : |
| index (UnitRefresh.orderedEnum A i)=i.val := by simp [index] |
| end MatchingFPRAS.TreeTyped.Numbered |
|
|
| end OAI |
| |
