File size: 2,371 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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.TreeFiniteOps
namespace OAI
namespace MatchingFPRAS.TreeTyped
open Polynomial
variable {A : Type} [Codec A]
def minNat : Realizer (fun x : ℕ×ℕ=>min x.1 x.2) :=
(choose leNat first second).congr (by intro x; simp only [decide_eq_true_eq]; exact (min_def x.1 x.2).symm)
lemma polynomial_minNat : PolynomialTime minNat :=
polynomial_congr _ (polynomial_choose polynomial_leNat polynomial_first polynomial_second)
def dropInit : Realizer (fun x : List A×ℕ=>(min x.2 x.1.length,x.1)) :=
pair (composition (pair second (composition first listLength)) minNat) first
lemma polynomial_dropInit : PolynomialTime (dropInit (A:=A)) := by
unfold dropInit
exact polynomial_pair (polynomial_composition (polynomial_pair polynomial_second
(polynomial_composition polynomial_first polynomial_listLength)) polynomial_minNat) polynomial_first
def dropFast : Realizer (fun x : List A×ℕ=>x.1.drop x.2) :=
(composition dropInit (iterateDown (composition second tail))).congr (by
intro x
change countDown dropStep (min x.2 x.1.length) x.1=x.1.drop x.2
rw [countDown_drop]
by_cases h : x.2≤x.1.length
· rw [min_eq_left h]
· rw [min_eq_right (by omega),List.drop_length,List.drop_eq_nil_of_le (by omega)])
lemma polynomial_dropFast : PolynomialTime (dropFast (A:=A)) := by
unfold dropFast
apply polynomial_congr
apply polynomial_boundedRepeat polynomial_dropInit
(polynomial_composition polynomial_second polynomial_tail) X X
(fun x _ z=>z.Sublist x.1)
· intro x
simp only [Polynomial.eval_X]
exact (min_le_right _ _).trans ((list_length_le_weight x.1).trans (by rw [weight_prod]; omega))
· intro x; exact List.Sublist.refl _
· intro x k hk z hz
exact (List.tail_sublist z).trans hz
· intro x k z hz
simp only [Polynomial.eval_X]
exact (weight_sublist hz).trans (by rw [weight_prod]; omega)
def getDFast (d : A) : Realizer (fun x : List A×ℕ=>x.1.getD x.2 d) :=
(composition dropFast (head d)).congr (by
intro x
simp only [Function.comp_apply, List.headD_eq_head?_getD, List.head?_drop,
List.getD_eq_getElem?_getD])
lemma polynomial_getDFast (d : A) : PolynomialTime (getDFast d) :=
polynomial_congr _ (polynomial_composition polynomial_dropFast (polynomial_head d))
end MatchingFPRAS.TreeTyped
end OAI
|