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namespace OAI
/-!
# Actual bipartite cell forests and their completion intervals
The graph vertices are the original row and column labels. Capacities
are attached to unordered row-column edges. The deletion-component cut
is constructed from the maximizing forest, so the hypotheses of the
physical cut estimate follow from actual graph properties.
-/
namespace ContingencyTables.ForestCapacity
open scoped BigOperators Classical
open SimpleGraph
open MaximumForest ForestCut
variable {I J : Type*} [Fintype I] [Fintype J]
def cellAdj (small : I → J → Bool) : I⊕J → I⊕J → Prop
| .inl i,.inr j => small i j=false
| .inr j,.inl i => small i j=false
| _,_ => False
def cellGraph (small : I → J → Bool) : SimpleGraph (I⊕J) where
Adj := cellAdj small
symm := ⟨by intro v w; cases v <;> cases w <;> exact id⟩
loopless := ⟨by intro v; cases v <;> exact id⟩
def pairCapacity (b : I → J → ℕ) : I⊕J → I⊕J → ℕ
| .inl i,.inr j => b i j
| .inr j,.inl i => b i j
| _,_ => 0
def capacity (b : I → J → ℕ) : Sym2 (I⊕J) → ℕ :=
Sym2.lift ⟨pairCapacity b,by intro v w; cases v <;> cases w <;> rfl⟩
omit [Fintype I] [Fintype J] in
@[simp] lemma capacity_cell (b : I → J → ℕ) (i : I) (j : J) :
capacity b s(Sum.inl i,Sum.inr j)=b i j := rfl
omit [Fintype I] [Fintype J] in
lemma cell_edge_injective : Function.Injective
(fun a : I×J => s(Sum.inl a.1,Sum.inr a.2)) := by
rintro ⟨i,j⟩ ⟨i',j'⟩ h
simpa only [Sym2.eq_iff,Prod.mk.injEq,Sum.inl.injEq,Sum.inr.injEq,
Sum.inl_ne_inr,Sum.inr_ne_inl,false_and,or_false] using h
noncomputable def treeCell (T : SimpleGraph (I⊕J)) (i : I) (j : J) : Bool :=
decide (T.Adj (.inl i) (.inr j))
/-- The actual maximum-forest cut gives the source's displacement and
interval bounds for every solution of the selected nonroot equations. -/
theorem completion_interval (small : I → J → Bool) (b : I → J → ℕ)
(T : SimpleGraph (I⊕J)) (hTG : T≤cellGraph small) (hT : T.IsAcyclic)
(hmax : ∀ F : SimpleGraph (I⊕J), F≤cellGraph small → F.IsAcyclic →
weight F (capacity b)≤weight T (capacity b))
(i₀ : I) (j₀ : J) (ht : T.Adj (.inl i₀) (.inr j₀)) (root : I⊕J)
(hroot : T.Reachable root (.inl i₀))
(x q z X : I → J → ℝ) (d : ℝ) (W : ℕ)
(hd : 1≤d) (hcard : (Fintype.card I*Fintype.card J:ℕ)≤d)
(hlarge : ∀ i j, small i j=false → W≤b i j ∧ 1≤b i j)
(hsmall : ∀ i j, small i j=true → b i j<W ∧
(0≤x i j ∧ x i j≤b i j) ∧ (0≤q i j ∧ q i j≤b i j))
(hX : ∀ i j, 0≤X i j ∧ X i j≤b i j)
(hfree : ∀ i j, small i j=false → treeCell T i j=false →
0≤z i j ∧ z i j≤b i j+1)
(hr : ∀ i, T.Reachable root (.inl i) → Sum.inl i≠root → rowDifference small x z X i=0)
(hc : ∀ j, T.Reachable root (.inr j) → Sum.inr j≠root → columnDifference small q z X j=0) :
-10*d*b i₀ j₀≤z i₀ j₀ ∧ z i₀ j₀≤10*d*b i₀ j₀ := by
let t : Sym2 (I⊕J) := s(Sum.inl i₀,Sum.inr j₀)
let D := T.deleteEdges {t}
obtain ⟨r,hrr,hrootSide,hcross⟩ := exists_root_avoiding_side T hT ht root hroot
let rowU : I → Bool := fun i => side D r (.inl i)
let colU : J → Bool := fun j => side D r (.inr j)
have hs₀ : small i₀ j₀=false := hTG ht
have hB : (1:ℝ)≤b i₀ j₀ := by exact_mod_cast (hlarge i₀ j₀ hs₀).2
have hdisp : |z i₀ j₀-X i₀ j₀|≤4*d*b i₀ j₀ := by
apply displacement_bound small (treeCell T) rowU colU x q z X
(fun i j => (b i j:ℝ)) i₀ j₀ d hcard hB hs₀ hcross
· intro i hi
change side D r (.inl i)=true at hi
apply hr i
· have hri : D.Reachable r (.inl i) := by
simpa only [side,decide_eq_true_eq] using hi
exact hrr.trans (hri.mono
(T.deleteEdges_le {t}))
· intro he
have hbad : side D r root=true := by simpa only [he] using hi
exact Bool.false_ne_true (hrootSide.symm.trans hbad)
· intro j hj
change side D r (.inr j)=true at hj
apply hc j
· have hrj : D.Reachable r (.inr j) := by
simpa only [side,decide_eq_true_eq] using hj
exact hrr.trans (hrj.mono
(T.deleteEdges_le {t}))
· intro he
have hbad : side D r root=true := by simpa only [he] using hj
exact Bool.false_ne_true (hrootSide.symm.trans hbad)
· exact hX
· intro i j hs
have hcap : b i j≤b i₀ j₀ := (Nat.le_of_lt (hsmall i j hs).1).trans
(hlarge i₀ j₀ hs₀).1
exact ⟨by exact_mod_cast hcap,(hsmall i j hs).2⟩
· intro i j htree hne
apply side_eq_of_tree_edge T t r
· exact of_decide_eq_true htree
· exact fun he => hne (cell_edge_injective he)
· intro i j hs htree
refine ⟨hfree i j hs htree,?_⟩
intro hcut
have hgraph : (cellGraph small).Adj (.inl i) (.inr j) := hs
have hcap := crossing_free_edge_le (cellGraph small) T (capacity b)
hTG hT hmax t (by simpa [t] using ht) r hgraph
(by simpa only [treeCell,decide_eq_false_iff_not] using htree) hcut
exact_mod_cast hcap
exact physical_interval_of_displacement d (b i₀ j₀) (X i₀ j₀) (z i₀ j₀)
hd (Nat.cast_nonneg _) (hX i₀ j₀) hdisp
end ContingencyTables.ForestCapacity
end OAI
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