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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 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 | import Mathlib
import OAI.Combinatorics.MatchingCount.Sampling.TreeOptionsMaps
import OAI.Combinatorics.MatchingCount.Machines.TreeCertify
import OAI.Combinatorics.MatchingCount.Machines.TreeColorRaw
import OAI.Combinatorics.MatchingCount.Graphs.TreeSwitchRaw
namespace OAI
namespace MatchingFPRAS.TreeTyped.SparseProgram
open TreeTyped ExecutableOptions
variable {E V C : Type} [Codec E] [Codec V] [Codec C] [LinearOrder E] [LinearOrder V]
attribute [local instance] Transport.Sparse.finsetOrder
abbrev TableInput (C E V : Type) := (C×(Finset V×Finset E))×Finset E
def colorTable (p : C×(E×V)→Bool) (q : C×(E×E)→Bool) (x : TableInput C E V) : Options (Finset E) :=
neighbors ((x.2,(certify p ((x.1.1,x.1.2.1),colorRaw q ((x.1.1,x.1.2.2),x.2))).erase x.2),x.1.2.1.card)
def colorTableRealizer {p : C×(E×V)→Bool} {q : C×(E×E)→Bool}
(P : Realizer p) (Q : Realizer q)
(Eq : Realizer (fun x : E×E=>decide (x.1=x.2)))
(Le : Realizer (fun x : E×E=>decide (x.1≤x.2))) : Realizer (colorTable p q) :=
(composition (pair (pair second (composition (pair second (composition
(pair (pair first.fst first.snd.fst) (composition (pair (pair first.fst first.snd.snd) second)
(colorRawRealizer Q Eq Le))) (certifyRealizer P))) (finsetErase (orderedSetEqual Eq))))
(composition first.snd.fst finsetCard)) neighborsRealizer).congr (by
intro x
apply congrArg neighbors
dsimp only [Function.comp_apply]
congr 2
ext b
simp only [Finset.mem_erase])
lemma polynomial_colorTable {p : C×(E×V)→Bool} {q : C×(E×E)→Bool}
{P : Realizer p} {Q : Realizer q}
{Eq : Realizer (fun x : E×E=>decide (x.1=x.2))}
{Le : Realizer (fun x : E×E=>decide (x.1≤x.2))}
(hP : PolynomialTime P) (hQ : PolynomialTime Q) (hE : PolynomialTime Eq) (hL : PolynomialTime Le) :
PolynomialTime (colorTableRealizer P Q Eq Le) := by
unfold colorTableRealizer Realizer.fst Realizer.snd
apply polynomial_congr
have := polynomial_colorRaw hQ hE hL
have := polynomial_certify hP
have := polynomial_orderedSetEqual hE
have := ExecutableOptions.polynomial_neighbors (A:=Finset E)
tree_cost
def degree (n : ℕ) : ℕ := 8*(n/2).choose 2
def degreeRealizer : Realizer degree := by
let half : Realizer (fun n : ℕ=>n/2) := composition (pair identity (constant 2)) quotient
let chooseTwo : Realizer (fun n : ℕ=>n.choose 2) :=
(composition (pair (composition (pair identity (composition (pair identity (constant 1)) subtract)) multiply)
(constant 2)) quotient).congr (fun n=>Nat.choose_two_right n |>.symm)
exact (composition (pair (constant 8) (composition half chooseTwo)) multiply).congr (fun _=>rfl)
lemma polynomial_degree : PolynomialTime degreeRealizer := by
unfold degreeRealizer
dsimp only
apply polynomial_congr
have := polynomial_quotient
tree_cost
def switchTable (p : C×(E×V)→Bool) (q : C×(E×(V×V))→Bool) (x : TableInput C E V) : Options (Finset E) :=
neighbors ((x.2,(certify p ((x.1.1,x.1.2.1),switchRaw q x)).erase x.2),degree x.1.2.1.card)
def switchTableRealizer {p : C×(E×V)→Bool} {q : C×(E×(V×V))→Bool}
(P : Realizer p) (Q : Realizer q)
(Eq : Realizer (fun x : E×E=>decide (x.1=x.2)))
(Le : Realizer (fun x : E×E=>decide (x.1≤x.2)))
(EqV : Realizer (fun x : V×V=>decide (x.1=x.2)))
(LeV : Realizer (fun x : V×V=>decide (x.1≤x.2))) : Realizer (switchTable p q) :=
(composition (pair (pair second (composition (pair second (composition
(pair (pair first.fst first.snd.fst) (switchRawRealizer Q Eq Le EqV LeV)) (certifyRealizer P)))
(finsetErase (orderedSetEqual Eq))))
(composition (composition first.snd.fst finsetCard) degreeRealizer)) neighborsRealizer).congr (by
intro x
apply congrArg neighbors
dsimp only [Function.comp_apply]
congr 2
ext b
simp only [Finset.mem_erase])
lemma polynomial_switchTable {p : C×(E×V)→Bool} {q : C×(E×(V×V))→Bool}
{P : Realizer p} {Q : Realizer q}
{Eq : Realizer (fun x : E×E=>decide (x.1=x.2))}
{Le : Realizer (fun x : E×E=>decide (x.1≤x.2))}
{EqV : Realizer (fun x : V×V=>decide (x.1=x.2))}
{LeV : Realizer (fun x : V×V=>decide (x.1≤x.2))}
(hP : PolynomialTime P) (hQ : PolynomialTime Q) (hE : PolynomialTime Eq) (hL : PolynomialTime Le)
(hV : PolynomialTime EqV) (hlV : PolynomialTime LeV) :
PolynomialTime (switchTableRealizer P Q Eq Le EqV LeV) := by
unfold switchTableRealizer Realizer.fst Realizer.snd
apply polynomial_congr
have := polynomial_switchRaw hQ hE hL hV hlV
have := polynomial_certify hP
have := polynomial_orderedSetEqual hE
have := polynomial_degree
have := ExecutableOptions.polynomial_neighbors (A:=Finset E)
tree_cost
variable [Fintype E] [Fintype V] (ends : E→Sym2 V)
attribute [local instance] Transport.Sparse.pmOrder
variable (hloop : ∀ e,¬(ends e).IsDiag) (hpar : ∀ e,(Finset.univ.filter (fun f=>ends f=ends e)).card≤2)
omit [Codec E] [Codec V] [Codec C] in
lemma colorTable_eq (p : C×(E×V)→Bool) (q : C×(E×E)→Bool) (c : C)
(hp : ∀ e v,p (c,(e,v))=decide (v∈ends e)) (hq : ∀ e f,q (c,(e,f))=decide (ends e=ends f))
(a : Transport.PM (fun e v=>v∈ends e)) :
colorTable p q ((c,(Finset.univ,Finset.univ)),a.val)=repr Subtype.val (Transport.Sparse.colorTable ends hloop hpar a) := by
unfold colorTable
have hraw : colorRaw q ((c,Finset.univ),a.val)=Transport.Sparse.colorRaw ends a.val := by
simp only [colorRaw,Transport.Sparse.colorRaw,hq,decide_eq_true_eq]
rw [hraw,certify_eq ends p c hp,Finset.card_univ]
have he:=Finset.map_erase (pmEmbedding ends).toEmbedding (Transport.Sparse.colors ends a) a
have hn:=neighbors_repr_map (pmEmbedding ends) a ((Transport.Sparse.colors ends a).erase a)
(Fintype.card V) (Transport.Sparse.color_degree ends hloop hpar a)
rw [he] at hn
exact hn
omit [Codec E] [Codec V] [Codec C] in
lemma switchTable_eq (p : C×(E×V)→Bool) (q : C×(E×(V×V))→Bool) (c : C)
(hp : ∀ e v,p (c,(e,v))=decide (v∈ends e)) (hq : ∀ e v w,q (c,(e,(v,w)))=decide (ends e=s(v,w)))
(a : Transport.PM (fun e v=>v∈ends e)) :
switchTable p q ((c,(Finset.univ,Finset.univ)),a.val)=repr Subtype.val (Transport.Sparse.switchTable ends hloop hpar a) := by
unfold switchTable
rw [switchRaw_eq ends q c hq,certify_eq ends p c hp,Finset.card_univ]
have he:=Finset.map_erase (pmEmbedding ends).toEmbedding (Transport.Sparse.switches ends a) a
have hn:=neighbors_repr_map (pmEmbedding ends) a ((Transport.Sparse.switches ends a).erase a)
(degree (Fintype.card V)) (Transport.Sparse.switch_degree ends hloop hpar a)
rw [he] at hn
exact hn
end MatchingFPRAS.TreeTyped.SparseProgram
end OAI
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