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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 | import OAI.Combinatorics.MatchingEntropy.EdgeReduction
namespace OAI
noncomputable section
open scoped BigOperators Topology
open Filter
universe uV uE uI uJ uA uB uC uW uF uD uι
namespace MatchingEntropyBounds
theorem separable_lower_aux (n : ℕ) :
∀ {V : Type uV} {E : Type uE} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]
(G : LooplessGraph V E), Fintype.card V=n → Even n →
∀ x ∈ G.polytope, separableLower x ≤ maxMatchingEntropy G x := by
induction n using Nat.strong_induction_on with
| h n ih =>
intro V E _ _ _ _ G hcard heven y hy
by_cases hn : n=0
· have : IsEmpty V := Fintype.card_eq_zero_iff.mp (hcard.trans hn)
have : IsEmpty E := ⟨fun e => isEmptyElim (G.left e)⟩
simpa [separableLower,entropy] using maxMatchingEntropy_nonneg G hy
· have heven' : Even (Fintype.card V) := hcard ▸ heven
have hPM : Nonempty G.Matching := by
obtain ⟨p,hp⟩ := (G.mem_polytope_iff y).mp hy
by_contra h
have : IsEmpty G.Matching := not_nonempty_iff.mp h
simpa using hp.1.2
obtain ⟨x,hx,hmax⟩ := separableDefect_attains_max G heven' hPM
obtain ⟨e₀,he₀⟩ := separable_max_has_forced G heven' (by omega) hx hmax
have hc := G.remaining_vertex_card e₀
have hlt : Fintype.card {v // G.RemainingVertex e₀ v}<n := by omega
have hev : Even (Fintype.card {v // G.RemainingVertex e₀ v}) := by
obtain ⟨r,hr⟩ := heven
refine ⟨r-1,?_⟩
omega
obtain ⟨p,hp⟩ := (G.mem_polytope_iff x).mp hx
have hs := G.forcedLaw_feasible hp e₀ he₀
have hxr := ((G.deleteEndpoints e₀).mem_polytope_iff _).mpr ⟨_,hs⟩
have hsmall := ih _ hlt (G.deleteEndpoints e₀) rfl hev _ hxr
rw [G.separableLower_forced hx e₀ he₀] at hsmall
have hxbound := hsmall.trans (G.maxMatchingEntropy_forced_le hx e₀ he₀)
have hcompare := hmax y hy
linarith
variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]
theorem separable_lower (G : LooplessGraph V E) (hPM : Nonempty G.Matching)
(x : E → ℝ) (hx : x ∈ G.polytope) : separableLower x ≤ maxMatchingEntropy G x := by
obtain ⟨M⟩ := hPM
have hev : Even (Fintype.card V) := ⟨M.val.card, by have hh := G.matching_card_twice M; omega⟩
exact separable_lower_aux (Fintype.card V) G rfl hev x hx
end MatchingEntropyBounds
end
end OAI
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