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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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Algorithm.TreeGeneratedStage
import OAI.Combinatorics.MatchingCount.Machines.TreeScaleStateUpdate
namespace OAI
namespace MatchingFPRAS.ExecutableSource
open TreeTyped TreeFraction TreeTyped.Numbered ExecutableBottleneck ExecutableHierarchy ExecutableRefresh Enlargement RealGeometry NumericLadder ExecutableLadder NumericMarking
attribute [local instance] NumberedHierarchy.labelOrder
attribute [local instance] Transport.sym2Order Transport.bagEdgeOrder Transport.bagEdgeFintype
attribute [local instance] UnitRefresh.Computable.allowedDec Transport.prodOrder Transport.sumOrder
local instance (priority := 2000) {A B : Type} [LinearOrder A] [LinearOrder B] : LE (A×B) := Transport.prodOrder.toLE
local instance (priority := 2000) {A B : Type} [LinearOrder A] [LinearOrder B] : LT (A×B) := Transport.prodOrder.toLT
local instance (priority := 2000) {A B : Type} [LinearOrder A] [LinearOrder B] : Preorder (A×B) := Transport.prodOrder.toPreorder
lemma logicalEdges_eq_ofFn (n : ℕ) : logicalEdges (List.range n)=
(List.ofFn (UnitRefresh.orderedEnum (LogicalEdge (Fin n)))).map logicalRepr := by
classical
rw [UnitRefresh.ofFn_orderedEnum]
have ha : (logicalEdges (List.range n)).Pairwise (· ≤ ·) :=
(logicalEdges_pairwise n).imp (fun h=>le_of_lt h)
have hb : (((Finset.univ : Finset (LogicalEdge (Fin n))).sort (· ≤ ·)).map logicalRepr).Pairwise (· ≤ ·) := by
apply List.pairwise_map.mpr
exact (Finset.pairwise_sort _ _).imp (fun h=>(logicalRepr_le _ _).mpr h)
have hn : (logicalEdges (List.range n)).Nodup := (logicalEdges_pairwise n).imp (fun h=>h.ne)
have hm := (Finset.sort_nodup (Finset.univ : Finset (LogicalEdge (Fin n))) (· ≤ ·)).map logicalRepr_injective
have hp : (logicalEdges (List.range n)).Perm
(((Finset.univ : Finset (LogicalEdge (Fin n))).sort (· ≤ ·)).map logicalRepr) := by
apply List.perm_of_nodup_nodup_toFinset_eq hn hm
ext e
simp only [List.mem_toFinset,List.mem_map,Finset.mem_sort,Finset.mem_univ,true_and]
constructor
· exact logicalEdges_complete e
· rintro ⟨a,rfl⟩; exact logicalRepr_mem a
exact hp.eq_of_pairwise (fun _ _ _ _ h₁ h₂=>@le_antisymm _ Transport.prodOrder.toPartialOrder _ _ h₁ h₂) ha hb
def centralIndexArg : Realizer (fun x : BagBuildInput×RawLogical=>(x.1,centralKey x)) :=
(TreeTyped.pair (first (A:=BagBuildInput) (B:=RawLogical)) centralKeyRealizer).congr (fun _=>rfl)
lemma polynomial_centralIndexArg : PolynomialTime centralIndexArg :=
polynomial_congr _ (TreeTyped.polynomial_pair polynomial_first polynomial_centralKey)
def centralIndex (x : BagBuildInput×RawLogical) : ℕ := tableIndex (x.1,centralKey x)
def centralIndexRealizer : Realizer centralIndex :=
(composition (f:=fun x : BagBuildInput×RawLogical=>(x.1,centralKey x)) (g:=tableIndex)
centralIndexArg tableIndexRealizer).congr (by intro x; simp only [Function.comp_apply,centralIndex])
lemma polynomial_centralIndex : PolynomialTime centralIndexRealizer :=
polynomial_congr _ (polynomial_composition polynomial_centralIndexArg polynomial_tableIndex)
def centralBinding (x : BagBuildInput×RawLogical) : RawLogical×ℕ :=
(x.2,centralIndex x)
def centralBindingRealizer : Realizer centralBinding :=
(TreeTyped.pair (second (A:=BagBuildInput) (B:=RawLogical)) centralIndexRealizer).congr (fun _=>rfl)
lemma polynomial_centralBinding : PolynomialTime centralBindingRealizer :=
polynomial_congr _ (TreeTyped.polynomial_pair polynomial_second polynomial_centralIndex)
def sourceBindings (x : BagBuildInput) : Bindings :=
(logicalEdges x.1.2).map (fun e=>centralBinding (x,e))
def sourceBindingsRealizer : Realizer sourceBindings := (rowsWith centralBindingRealizer
(composition (first (A:=BuildInput) (B:=List Bool)).snd logicalEdgesRealizer)).congr (fun _=>rfl)
lemma polynomial_sourceBindings : PolynomialTime sourceBindingsRealizer :=
polynomial_congr _ (polynomial_rowsWith polynomial_centralBinding
(polynomial_composition (polynomial_snd polynomial_first) polynomial_logicalEdges))
lemma sourceBindings_source {n : ℕ} [Nontrivial (Fin n)] (hn : 2≤n)
(w : Fin n→Fin n→ℚ) (hw : ∀ i j,0≤w i j) (K : ℕ) :
sourceBindings (((matrixRep w,List.replicate (2*K+2) false),List.range n),List.replicate (2*K) false)=
(List.ofFn (UnitRefresh.orderedEnum (LogicalEdge (Fin n)))).map
(fun e=>(logicalRepr e,index (RationalSource.centralEdge w K e))) := by
unfold sourceBindings
rw [logicalEdges_eq_ofFn,List.map_map]
apply List.map_congr_left
intro e _he
dsimp only [Function.comp_def,centralBinding,centralIndex]
rw [centralKey_source hn w hw K]
unfold tableIndex
dsimp only
rw [sourceTables_catalog hn w hw K,sourceCatalog_index]
abbrev GeneratedUpdateConfig := (BagBuildInput×ℕ)×ExecutableScaling.Vector
def generatedUpdateConfig (x : GeneratedUpdateConfig) : UpdateConfig :=
((x.1.2,x.1.1.1.2),(sourceBindings x.1.1,x.2))
def generatedUpdateConfigRealizer : Realizer generatedUpdateConfig := (pair
(pair first.snd first.fst.fst.snd)
(pair (composition first.fst sourceBindingsRealizer) second)).congr (fun _=>rfl)
lemma polynomial_generatedUpdateConfig : PolynomialTime generatedUpdateConfigRealizer := by
unfold generatedUpdateConfigRealizer
apply polynomial_congr
dsimp only [Realizer.fst, Realizer.snd]
have := polynomial_sourceBindings
tree_cost
lemma generatedUpdateConfig_source {G : GraphInput} [Nontrivial (Fin G.n)] (hn : 2≤G.n)
{j : ℕ} (s : FiniteTrial.ScaleState G j) (K : ℕ) :
generatedUpdateConfig (((((matrixRep s.activity,List.replicate (2*K+2) false),List.range G.n),
List.replicate (2*K) false),FiniteTrial.sourceD (G:=G) K),ExecutableScaling.vectorRep s.scale)=
sourceUpdateConfig s K := by
unfold generatedUpdateConfig sourceUpdateConfig
dsimp only
rw [sourceBindings_source hn s.activity (fun i j=>(s.activity_pos i j).le) K]
end MatchingFPRAS.ExecutableSource
end OAI
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