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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 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | import Mathlib
import OAI.Combinatorics.MatchingCount.Algorithm.OnlineInitial
import OAI.Combinatorics.MatchingCount.Complexity.OnlineScheduleCost
namespace OAI
namespace MatchingFPRAS.OnlineInitial
open TreeTyped TreeFraction OnlineSchedule Polynomial NumericBounds
lemma work_root : WorkBound rootRealizer id := by
unfold rootRealizer
apply work_then_polynomial _ polynomial_findRoot
apply work_pair _ _ (4*X+1) (fun n=>by simpa using weight_nat_numeric n)
· exact work_of_polynomial (polynomial_composition (polynomial_pair polynomial_identity (polynomial_constant _))
polynomial_subtract) (4*X+1) (fun n=>by simpa using weight_nat_numeric n)
· exact work_composition (work_of_polynomial polynomial_identity (4*X+1)
(fun n=>by simpa using weight_nat_numeric n)) fuel_numeric_work X (by simp)
lemma work_initialScale : WorkBound initialScaleRealizer id := by
unfold initialScaleRealizer
apply work_then_polynomial _ polynomial_div
apply work_pair _ _ (4*X+1) (fun n=>by simpa using weight_nat_numeric n)
· exact work_of_polynomial (polynomial_constant _) (4*X+1) (fun n=>by simpa using weight_nat_numeric n)
· exact work_pair work_root (work_of_polynomial (polynomial_constant _) (4*X+1)
(fun n=>by simpa using weight_nat_numeric n)) (4*X+1) (fun n=>by simpa using weight_nat_numeric n)
lemma polynomial_scaleVector : PolynomialTime scaleVectorRealizer := by
unfold scaleVectorRealizer
apply polynomial_congr
apply work_then_polynomial _ (polynomial_mapWith (polynomial_pair polynomial_second polynomial_first))
apply work_pair _ polynomial_identity X (by simp)
exact work_composition polynomial_listLength work_initialScale X (fun vs=>by simpa using list_length_le_weight vs)
lemma polynomial_count : PolynomialTime countRealizer := by
unfold countRealizer nRealizer Realizer.fst Realizer.snd
apply polynomial_composition _ polynomial_sampleCount
tree_cost
lemma polynomial_seedD : PolynomialTime seedDRealizer := by
unfold seedDRealizer Realizer.fst
apply polynomial_pair polynomial_first
apply polynomial_pair polynomial_second
apply polynomial_composition _ polynomial_distortion
tree_cost
def configNumeric (x : Config) : ℕ := x.1.1.1.1.length+x.1.2+x.2.1+x.2.2+
ladderSchedule (x.2,x.1.2)+Transport.ladderQ x.2.1 x.2.2
def configSize (x : Config) : ℕ := configNumeric x+weight x
def seedSize (x : Seed) : ℕ := x.1.1.length+natCeil (inverse x.2.1)+configNumeric (config x)+weight x
lemma weight_configSize (x : Config) : weight x ≤ configSize x := by unfold configSize;omega
lemma weight_seedSize (x : Seed) : weight x ≤ seedSize x := by unfold seedSize;omega
lemma config_fields (x : Config) : x.1.1.1.1.length ≤ configSize x ∧ x.1.2 ≤ configSize x ∧
x.2.1 ≤ configSize x ∧ x.2.2 ≤ configSize x ∧
ladderSchedule (x.2,x.1.2) ≤ configSize x ∧Transport.ladderQ x.2.1 x.2.2 ≤ configSize x := by
unfold configSize configNumeric;omega
lemma seed_fields (x : Seed) : x.1.1.length ≤ seedSize x ∧ natCeil (inverse x.2.1) ≤ seedSize x ∧
coolingSchedule (x.1.1.length,x.2.1) ≤ seedSize x ∧configNumeric (config x) ≤ seedSize x := by
unfold seedSize configNumeric config seedD seedN seedK
dsimp only
omega
lemma work_seedK : WorkBound seedKRealizer seedSize := by
have lift {A : Type} [Codec A] {f : Seed→A} {F : Realizer f} (hF : PolynomialTime F) :
WorkBound F seedSize := work_of_polynomial hF X (fun x=>by simpa using weight_seedSize x)
unfold seedKRealizer
apply work_congr
apply work_pair (lift polynomial_identity) _ X (fun x=>by simpa using weight_seedSize x)
apply work_comp_bound (lift (polynomial_pair (polynomial_composition (polynomial_fst polynomial_first)
polynomial_listLength) (polynomial_fst polynomial_second))) work_coolingSchedule
have hw:=NumericBounds.weight (lift (polynomial_pair (polynomial_composition (polynomial_fst polynomial_first)
polynomial_listLength) (polynomial_fst polynomial_second)))
exact add (add (of_le (fun x=>(seed_fields x).1)) (of_le (fun x=>(seed_fields x).2.1))) hw
def seedNSize (x : Seed×ℕ) : ℕ := x.1.1.1.length+x.2+weight x
lemma work_seedN : WorkBound seedNRealizer seedNSize := by
have lift {A : Type} [Codec A] {f : (Seed×ℕ)→A} {F : Realizer f} (hF : PolynomialTime F) :
WorkBound F seedNSize := work_of_polynomial hF X (fun x=>by simp [seedNSize])
unfold seedNRealizer
apply work_congr
apply work_pair (lift polynomial_identity) _ X (fun x=>by simp [seedNSize])
apply work_comp_bound (lift (polynomial_pair (polynomial_composition (polynomial_fst (polynomial_fst polynomial_first))
polynomial_listLength) polynomial_second)) work_vertexNumber
have hw:=NumericBounds.weight (lift (polynomial_pair (polynomial_composition (polynomial_fst (polynomial_fst polynomial_first))
polynomial_listLength) polynomial_second))
exact add (add (of_le (fun x=>by dsimp only [Function.comp_apply]; unfold seedNSize;omega)) (of_le (fun x=>by dsimp only [Function.comp_apply]; unfold seedNSize;omega))) hw
lemma work_config : WorkBound configRealizer seedSize := by
unfold configRealizer
apply work_then_polynomial _ polynomial_seedD
apply work_comp_bound work_seedK work_seedN
have hw:=NumericBounds.weight work_seedK
exact add (add (of_le (fun x=>(seed_fields x).1)) (of_le (fun x=>(seed_fields x).2.2.1))) hw
lemma bound_configSize : Bound (configSize ∘ config) seedSize :=
add (of_le (fun x=>(seed_fields x).2.2.2)) (NumericBounds.weight work_config)
end MatchingFPRAS.OnlineInitial
end OAI
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