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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 | 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
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