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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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Graphs.TreeBagCorrect
import OAI.Combinatorics.MatchingCount.Graphs.TreeHierarchyReindex
namespace OAI
namespace MatchingFPRAS.ExecutableHierarchy
open TreeTyped TreeFraction TreeTyped.Numbered ExecutableBottleneck NumberedHierarchy
variable {V : Type} [Fintype V] [LinearOrder V]
variable (B : V→V→ℚ) (hB : ∀ a b,0 ≤ B a b)
(hsym : ∀ a b,B a b=B b a)
(htri : ∀ x y z,x≠y → x≠z → y≠z → min (B x z) (B z y)≤B x y)
attribute [local instance] NumberedHierarchy.labelOrder
local instance (priority := 2000) treeBagsLabelLE {U : Type} [Fintype U] [LinearOrder U] {H : ℕ}
(C : U→U→ℚ) : LE (Label C H) := (labelOrder C H).toLE
variable (xs : List Bool) {E : Type} [Fintype E] [DecidableEq E] (left right : E→V)
(es : List (ℕ×ℕ))
(hes : ∀ p,p∈es ↔ ∃ e,(index (left e),index (right e))=p)
include hB hes in
lemma arbitrary_bag_members (v : V) (i : ℕ) :
i∈bag (((matrixRep (reindex B (orderedIso V)),xs),es),index v) ↔
∃ e,(v=left e ∨ v=right e) ∧ index (edgeLabel B hsym htri left right xs.length e)=i := by
let f:=orderedIso V
have hh:=bag_members (reindex B f) (fun a b=>hB (f a) (f b))
(reindex_symm B f hsym) (reindex_tri B f htri) xs (f.symm ∘ left) (f.symm ∘ right)
es hes (f.symm v) i
change i∈bag (((matrixRep (reindex B f),xs),es),index v) ↔ _ at hh
rw [hh]
apply exists_congr
intro a
have he:=labelReindex_index B f xs.length (edgeLabel (reindex B f)
(reindex_symm B f hsym) (reindex_tri B f htri)
(f.symm ∘ left) (f.symm ∘ right) xs.length a)
rw [edgeLabel_reindex B f hsym htri] at he
change (f.symm v=f.symm (left a) ∨ f.symm v=f.symm (right a)) ∧ _ ↔ _
simp only [f.symm.injective.eq_iff]
simpa only [index] using iff_of_eq (congrArg (fun j=>(v=left a ∨ v=right a) ∧ j=i) he.symm)
noncomputable def arbitraryNumberedBags (v : V) :
Finset (Fin (Fintype.card (Label B xs.length))) :=
(NumberedHierarchy.bags B hsym htri left right xs.length v).map
(UnitRefresh.orderedEnum (Label B xs.length)).symm.toEmbedding
include hB hes in
lemma arbitrary_bag_perm (v : V) :
(bag (((matrixRep (reindex B (orderedIso V)),xs),es),index v)).Perm
(ExecutableDP.bagRep (arbitraryNumberedBags B hsym htri xs left right v)) := by
apply (List.perm_ext_iff_of_nodup (List.nodup_dedup _) ((Finset.nodup_toList _).map Fin.val_injective)).mpr
intro i
change i∈bag (((matrixRep (reindex B (orderedIso V)),xs),es),index v) ↔ _
rw [arbitrary_bag_members B hB hsym htri xs left right es hes]
simp only [List.mem_map,Finset.mem_toList,arbitraryNumberedBags,
Finset.mem_map,Equiv.toEmbedding_apply,NumberedHierarchy.mem_bags]
constructor
· rintro ⟨e,he,hi⟩
exact ⟨_,⟨_,⟨e,he,rfl⟩,rfl⟩,hi⟩
· rintro ⟨a,⟨l,⟨e,he,hel⟩,hla⟩,hai⟩
subst l
subst a
exact ⟨e,he,hai⟩
include hB hes in
lemma arbitrary_bagTable :
List.Forall₂ List.Perm (bagTable ((matrixRep (reindex B (orderedIso V)),xs),es))
(List.ofFn (fun i=>ExecutableDP.bagRep
(arbitraryNumberedBags B hsym htri xs left right (UnitRefresh.orderedEnum V i)))) := by
unfold bagTable
rw [matrixRep_length,ExecutableDP.map_range_ofFn]
apply List.forall₂_iff_get.mpr
refine ⟨by simp,?_⟩
intro i hi _right_bound
have hh:=arbitrary_bag_perm B hB hsym htri xs left right es hes
(UnitRefresh.orderedEnum V ⟨i,by simpa using hi⟩)
simpa only [List.get_eq_getElem,List.getElem_ofFn,index,Equiv.symm_apply_apply] using hh
include hB hes in
lemma arbitrary_bagTable_length (H : ℕ) (h : xs.length=H) :
List.Forall₂ List.Perm (bagTable ((matrixRep (reindex B (orderedIso V)),xs),es))
(List.ofFn (fun i=>ExecutableDP.bagRep
((NumberedHierarchy.bags B hsym htri left right H (UnitRefresh.orderedEnum V i)).map
(UnitRefresh.orderedEnum (Label B H)).symm.toEmbedding))) := by
subst H
exact arbitrary_bagTable B hB hsym htri xs left right es hes
include hB hes in
lemma arbitrary_bagTable_numbered (H : ℕ) (h : xs.length=H) :
List.Forall₂ List.Perm (bagTable ((matrixRep (reindex B (orderedIso V)),xs),es))
(ExecutableDP.orderedRep (UnitRefresh.numberedBags
(NumberedHierarchy.bags B hsym htri left right H))) :=
arbitrary_bagTable_length B hB hsym htri xs left right es hes H h
include hB hes in
lemma arbitrary_bagTable_enumerated (H : ℕ) (h : xs.length=H) :
List.Forall₂ List.Perm
(bagTable ((matrixRep (fun i j=>B (UnitRefresh.orderedEnum V i) (UnitRefresh.orderedEnum V j)),xs),es))
(ExecutableDP.orderedRep (UnitRefresh.numberedBags
(NumberedHierarchy.bags B hsym htri left right H))) :=
arbitrary_bagTable_numbered B hB hsym htri xs left right es hes H h
@[instance_reducible] def ordinaryLabelFintype (H : ℕ) : Fintype (Label B H) := inferInstance
include hB hes in
lemma arbitrary_bagTable_fintype (H : ℕ) (h : xs.length=H)
(F : Fintype (Label B H)) :
letI := F
List.Forall₂ List.Perm
(bagTable ((matrixRep (fun i j=>B (UnitRefresh.orderedEnum V i) (UnitRefresh.orderedEnum V j)),xs),es))
(ExecutableDP.orderedRep (UnitRefresh.numberedBags
(NumberedHierarchy.bags B hsym htri left right H))) := by
have hf : F=ordinaryLabelFintype B H := Subsingleton.elim _ _
subst F
exact arbitrary_bagTable_enumerated B hB hsym htri xs left right es hes H h
end MatchingFPRAS.ExecutableHierarchy
end OAI
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