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import OAI.Combinatorics.MatchingCount.Graphs.Edge2
namespace OAI
namespace MatchingFPRAS.UnitRefresh
open Finset
variable {V W L : Type*} [Fintype V] [Fintype W] [Fintype L]
[DecidableEq V] [DecidableEq W] [DecidableEq L]
noncomputable def assignmentEquiv (b : V → Finset L) (c : W → Finset L)
(e : V ≃ W) (he : ∀ x,c (e x)=b x) : Assignment b ≃ Assignment c where
toFun f := ⟨fun y => f.val (e.symm y),fun y => by
have hy : c y=b (e.symm y) := by simpa using he (e.symm y)
rw [hy]; exact f.property (e.symm y)⟩
invFun f := ⟨fun x => f.val (e x),fun x => by rw [←he]; exact f.property _⟩
left_inv f := Subtype.ext (by funext x; simp)
right_inv f := Subtype.ext (by funext x; simp)
omit [Fintype L] [DecidableEq V] [DecidableEq W] in
lemma fiberCount_equiv [Fintype L] [DecidableEq V] [DecidableEq W]
(e : V ≃ W) (f : V → L) (g : W → L)
(h : ∀ x,g (e x)=f x) (l : L) : fiberCount f l=fiberCount g l := by
unfold fiberCount
exact Fintype.card_congr (e.subtypeEquiv (fun x => by rw [h]))
lemma assignmentPartition_equiv (b : V → Finset L) (c : W → Finset L)
(e : V ≃ W) (he : ∀ x,c (e x)=b x) (w : L → ℕ → ℕ) :
assignmentPartition b w = assignmentPartition c w := by
classical
unfold assignmentPartition
apply Fintype.sum_equiv (assignmentEquiv b c e he)
intro f
apply prod_congr rfl
intro l hl
congr 1
apply fiberCount_equiv e
intro x
simp [assignmentEquiv]
abbrev Private (b : V → Finset L) (l : L) := {x : V // b x={l}}
abbrev Shared (b : V → Finset L) (l : L) := {x : V // l∈b x ∧ b x≠{l}}
abbrev Outside (b : V → Finset L) (l : L) := {x : V // l∉b x}
noncomputable def splitLeaf (b : V → Finset L) (l : L) :
V ≃ Private b l ⊕ Shared b l ⊕ Outside b l where
toFun x := if h : b x={l} then .inl ⟨x,h⟩ else
if h' : l∈b x then .inr (.inl ⟨x,h',h⟩) else .inr (.inr ⟨x,h'⟩)
invFun := Sum.elim Subtype.val (Sum.elim Subtype.val Subtype.val)
left_inv x := by dsimp only; split_ifs <;> rfl
right_inv x := by
rcases x with x | (x | x)
· simp [x.property]
· simp [x.property.1,x.property.2]
· have hx : b x.val≠{l} := by intro h; exact x.property (by simp [h])
simp [hx,x.property]
def LeafAt (b : V → Finset L) (l p : L) : Prop :=
∀ x,l∈b x → b x={l} ∨ b x={l,p}
omit [Fintype V] [Fintype L] [DecidableEq V] in
lemma splitLeaf_bags [Fintype V] [Fintype L] [DecidableEq V]
(b : V → Finset L) (l p : L) (h : LeafAt b l p) (x : V) :
leafBags l p (fun y : Outside b l => b y.val) (splitLeaf b l x)=b x := by
classical
unfold splitLeaf
simp only [Equiv.coe_fn_mk]
split_ifs with he hm
· exact he.symm
· exact ((h x hm).resolve_left he).symm
· rfl
theorem tree_leaf_elimination (b : V → Finset L) (l p : L) (hlp : l≠p)
(h : LeafAt b l p) (w : L → ℕ → ℕ) :
assignmentPartition b w =
assignmentPartition (fun x : Outside b l => b x.val)
(leafUpdate (Fintype.card (Private b l)) (Fintype.card (Shared b l)) l p w) := by
classical
rw [assignmentPartition_equiv b _ (splitLeaf b l) (splitLeaf_bags b l p h)]
exact leaf_elimination l p hlp _ (fun x => x.property) w
def rootBags {U : Type*} (l : L) (b : W → Finset L) : U ⊕ W → Finset L :=
Sum.elim (fun _ => {l}) b
noncomputable def rootAssignmentEquiv {U : Type*} [Fintype U] [DecidableEq U]
(l : L) (b : W → Finset L) : Assignment (rootBags (U:=U) l b) ≃ Assignment b where
toFun f := ⟨fun x => f.val (.inr x),fun x => f.property _⟩
invFun f := ⟨Sum.elim (fun _ => l) f.val,by
intro x; rcases x with x | x
· simp [rootBags]
· exact f.property x⟩
left_inv f := by
apply Subtype.ext
funext x
rcases x with x | x
· have h := f.property (.inl x)
have hx : f.val (.inl x)=l := by simpa [rootBags] using h
exact hx.symm
· rfl
right_inv f := rfl
noncomputable def rootUpdate (u : ℕ) (l : L) (w : L → ℕ → ℕ) : L → ℕ → ℕ :=
fun a r => if a=l then w l u else w a r
lemma root_elimination {U : Type*} [Fintype U] [DecidableEq U]
(l : L) (b : W → Finset L) (hb : ∀ x,l∉b x) (w : L → ℕ → ℕ) :
assignmentPartition (rootBags (U:=U) l b) w =
assignmentPartition b (rootUpdate (Fintype.card U) l w) := by
classical
unfold assignmentPartition
apply Fintype.sum_equiv (rootAssignmentEquiv l b)
intro f
have hl : fiberCount f.val l = Fintype.card U := by
rw [fiberCount_sum]
have he : (fun x : U => f.val (.inl x))=(fun _ => l) := by
funext x; simpa [rootBags] using f.property (.inl x)
rw [he,fiberCount_const,ite_eq_left rfl]
have hz : fiberCount (fun x : W => f.val (.inr x)) l=0 := by
apply fiberCount_zero
intro x hx
exact hb x (hx ▸ f.property (.inr x))
omega
apply prod_congr rfl
intro a ha
by_cases h : a=l
· subst a; simp [rootUpdate,hl]
· simp only [rootUpdate,ite_eq_right h]
congr 1
rw [fiberCount_sum]
have he : (fun x : U => f.val (.inl x))=(fun _ => l) := by
funext x; simpa [rootBags] using f.property (.inl x)
rw [he,fiberCount_const,ite_eq_right (Ne.symm h),zero_add]
rfl
noncomputable def splitRoot (b : V → Finset L) (l : L) :
V ≃ {x : V // l∈b x} ⊕ Outside b l where
toFun x := if h : l∈b x then .inl ⟨x,h⟩ else .inr ⟨x,h⟩
invFun := Sum.elim Subtype.val Subtype.val
left_inv x := by dsimp only; split_ifs <;> rfl
right_inv x := by rcases x with x | x <;> simp [x.property]
theorem tree_root_elimination (b : V → Finset L) (l : L)
(h : ∀ x,l∈b x → b x={l}) (w : L → ℕ → ℕ) :
assignmentPartition b w =
assignmentPartition (fun x : Outside b l => b x.val)
(rootUpdate (Fintype.card {x : V // l∈b x}) l w) := by
classical
have he (x : V) : rootBags l (fun y : Outside b l => b y.val) (splitRoot b l x)=b x := by
unfold splitRoot
simp only [Equiv.coe_fn_mk]
split_ifs with hx
· exact (h x hx).symm
· rfl
rw [assignmentPartition_equiv b _ (splitRoot b l) he]
exact root_elimination l _ (fun x => x.property) w
open Finset
variable {V L : Type*} [Fintype V] [Fintype L] [DecidableEq V] [DecidableEq L]
noncomputable def activePartition (b : V → Finset L) (R : Finset V) (w : L → ℕ → ℕ) : ℕ :=
assignmentPartition (fun x : R => b x.val) w
def trim (b : V → Finset L) (R : Finset V) (l : L) : Finset V := R.filter fun x => l∉b x
def privateCount (b : V → Finset L) (R : Finset V) (l : L) : ℕ :=
(R.filter fun x => b x={l}).card
def sharedCount (b : V → Finset L) (R : Finset V) (l : L) : ℕ :=
(R.filter fun x => l∈b x ∧ b x≠{l}).card
noncomputable def activeFilterEquiv (R : Finset V) (P : V → Prop) [DecidablePred P] :
{x : R // P x.val} ≃ (R.filter P) where
toFun x := ⟨x.val.val,mem_filter.mpr ⟨x.val.property,x.property⟩⟩
invFun x := ⟨⟨x.val,(mem_filter.mp x.property).1⟩,(mem_filter.mp x.property).2⟩
left_inv _ := rfl
right_inv _ := rfl
omit [Fintype V] [DecidableEq V] in
lemma active_card_filter [Fintype V] [DecidableEq V]
(R : Finset V) (P : V → Prop) [DecidablePred P] :
Fintype.card {x : R // P x.val}=(R.filter P).card := by
rw [Fintype.card_congr (activeFilterEquiv R P),Fintype.card_coe]
omit [Fintype V] in
lemma active_outside [Fintype V]
(b : V → Finset L) (R : Finset V) (l : L) (w : L → ℕ → ℕ) :
assignmentPartition (fun x : Outside (fun y : R => b y.val) l => b x.val.val) w =
activePartition b (trim b R l) w := by
change assignmentPartition (fun x : {y : R // l∉b y.val} => b x.val.val) w =
assignmentPartition (fun x : R.filter (fun y => l∉b y) => b x.val) w
apply assignmentPartition_equiv _ _ (activeFilterEquiv R (fun x => l∉b x))
intro x
rfl
theorem active_leaf_elimination (b : V → Finset L) (R : Finset V) (l p : L) (hlp : l≠p)
(h : LeafAt (fun x : R => b x.val) l p) (w : L → ℕ → ℕ) :
activePartition b R w = activePartition b (trim b R l)
(leafUpdate (privateCount b R l) (sharedCount b R l) l p w) := by
unfold activePartition
rw [tree_leaf_elimination _ l p hlp h,active_outside]
rw [show Fintype.card (Private (fun x : R => b x.val) l)=privateCount b R l from
active_card_filter R (fun x => b x={l}),
show Fintype.card (Shared (fun x : R => b x.val) l)=sharedCount b R l from
active_card_filter R (fun x => l∈b x ∧ b x≠{l})]
rfl
lemma active_root_elimination (b : V → Finset L) (R : Finset V) (l : L)
(h : ∀ x : R,l∈b x.val → b x.val={l}) (w : L → ℕ → ℕ) :
activePartition b R w = activePartition b (trim b R l)
(rootUpdate ((R.filter fun x => l∈b x).card) l w) := by
unfold activePartition
rw [tree_root_elimination _ l h,active_outside,active_card_filter R (fun x => l∈b x)]
rfl
lemma partition_isEmpty {W : Type*} [Fintype W] [DecidableEq W] [IsEmpty W]
(b : W → Finset L) (w : L → ℕ → ℕ) : assignmentPartition b w=∏ l,w l 0 := by
classical
let : Unique (Assignment b) :=
{ default := ⟨isEmptyElim,fun x => isEmptyElim x⟩
uniq := fun f => Subtype.ext (by funext x; exact isEmptyElim x) }
simp [assignmentPartition,fiberCount]
omit [Fintype V] in
lemma active_empty [Fintype V] (b : V → Finset L) (w : L → ℕ → ℕ) :
activePartition b ∅ w=∏ l,w l 0 := partition_isEmpty _ _
def RankedBags (b : V → Finset L) (rank : L → ℕ) (parent : L → Option L) : Prop :=
∀ x a,a ∈ b x → ∀ a'∈b x,a≠a' →
(rank a<rank a' ∧ parent a'=some a) ∨ (rank a'<rank a ∧ parent a=some a')
noncomputable def advanceTable (b : V → Finset L) (R : Finset V) (parent : L → Option L)
(l : L) (w : L → ℕ → ℕ) : L → ℕ → ℕ :=
match parent l with
| some p => leafUpdate (privateCount b R l) (sharedCount b R l) l p w
| none => rootUpdate ((R.filter fun x => l∈b x).card) l w
noncomputable def tableDP (b : V → Finset L) (parent : L → Option L) :
List L → Finset V → (L → ℕ → ℕ) → (L → ℕ → ℕ)
| [],_,w => w
| l::ls,R,w => tableDP b parent ls (trim b R l) (advanceTable b R parent l w)
def LegalPlan (b : V → Finset L) (parent : L → Option L) : List L → Finset V → Prop
| [],R => R=∅
| l::ls,R =>
(match parent l with
| some p => l≠p ∧ LeafAt (fun x : R => b x.val) l p
| none => ∀ x : R,l∈b x.val → b x.val={l}) ∧
LegalPlan b parent ls (trim b R l)
theorem tableDP_correct (b : V → Finset L) (parent : L → Option L)
(ls : List L) (R : Finset V) (hp : LegalPlan b parent ls R) (w : L → ℕ → ℕ) :
activePartition b R w=∏ l,tableDP b parent ls R w l 0 := by
induction ls generalizing R w with
| nil =>
have hR : R=∅ := hp
subst R
exact active_empty b w
| cons l ls ih =>
have ht : activePartition b R w =
activePartition b (trim b R l) (advanceTable b R parent l w) := by
cases hpar : parent l with
| none =>
unfold advanceTable; rw [hpar]
exact active_root_elimination b R l (by simpa only [LegalPlan,hpar] using hp.1) w
| some p =>
have h := hp.1
simp only [hpar] at h
unfold advanceTable; rw [hpar]
exact active_leaf_elimination b R l p h.1 h.2 w
rw [ht]
exact ih _ hp.2 _
omit [Fintype V] [Fintype L] [DecidableEq V] in
theorem rank_order_legal [Fintype V] [Fintype L] [DecidableEq V]
(b : V → Finset L) (rank : L → ℕ) (parent : L → Option L)
(hb : RankedBags b rank parent)
(hp : ∀ l p,parent l=some p → rank p<rank l)
(ls : List L) (hs : ls.Pairwise (fun a a' => rank a'≤rank a))
(R : Finset V) (hne : ∀ x∈R,(b x).Nonempty)
(hc : ∀ x∈R,∀ a∈b x,a∈ls) : LegalPlan b parent ls R := by
induction ls generalizing R with
| nil =>
apply eq_empty_iff_forall_notMem.mpr
intro x hx
obtain ⟨a,ha⟩ := hne x hx
exact List.not_mem_nil (hc x hx a ha)
| cons l ls ih =>
have hs' := List.pairwise_cons.mp hs
have hmax (x : R) (a : L) (ha : a∈b x.val) : rank a≤rank l := by
have h := hc x.val x.property a ha
rcases List.mem_cons.mp h with h | h
· exact h ▸ le_rfl
· exact hs'.1 a h
have hparent (x : R) (hl : l∈b x.val) (a : L) (ha : a∈b x.val) (hal : a≠l) :
parent l=some a := by
rcases hb x.val l hl a ha (Ne.symm hal) with h | h
· exact (Nat.not_lt_of_ge (hmax x a ha) h.1).elim
· exact h.2
constructor
· cases hpar : parent l with
| none =>
intro x hl
apply eq_singleton_iff_unique_mem.mpr
exact ⟨hl,fun a ha => by
by_contra hal
have h := hparent x hl a ha hal
rw [hpar] at h
cases h⟩
| some p =>
constructor
· intro h; subst p
exact Nat.lt_irrefl _ (hp l l hpar)
· intro x hl
by_cases h : b x.val={l}
· exact Or.inl h
· right
apply Subset.antisymm
· intro a ha
by_cases hal : a=l
· simp [hal]
· have hpa : p=a := Option.some.inj (hpar.symm.trans (hparent x hl a ha hal))
simp [hpa]
· intro a ha
simp only [mem_insert,mem_singleton] at ha
rcases ha with ha | ha
· simpa [ha] using hl
· rw [ha]
by_contra hpn
apply h
apply eq_singleton_iff_unique_mem.mpr
refine ⟨hl,fun a ha => ?_⟩
by_contra hal
have hpa : p=a := Option.some.inj (hpar.symm.trans (hparent x hl a ha hal))
exact hpn (hpa ▸ ha)
· apply ih hs'.2
· intro x hx
exact hne x (mem_filter.mp hx).1
· intro x hx a ha
have hx' := mem_filter.mp hx
have h := hc x hx'.1 a ha
rcases List.mem_cons.mp h with hh | hh
· exact (hx'.2 (hh ▸ ha)).elim
· exact hh
end MatchingFPRAS.UnitRefresh
end OAI
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