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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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Sampling.DiscreteLadderSteps
namespace OAI
namespace MatchingFPRAS.UnitRefresh
lemma orderedEnum_lt {A : Type} [Fintype A] [LinearOrder A]
(i j : Fin (Fintype.card A)) : orderedEnum A i < orderedEnum A j ↔ i < j := by
change ((Finset.univ : Finset A).orderIsoOfFin (Finset.card_univ) i).val <
((Finset.univ : Finset A).orderIsoOfFin (Finset.card_univ) j).val ↔ _
exact ((Finset.univ : Finset A).orderIsoOfFin (Finset.card_univ)).lt_iff_lt
end MatchingFPRAS.UnitRefresh
namespace MatchingFPRAS.BottleneckAlgorithm
open Finset
variable {V W α : Type} [Fintype V] [Fintype W] [LinearOrder V] [LinearOrder W]
[LinearOrder α] [OfNat α 1]
omit [Fintype V] [Fintype W] in
lemma seed_reindex (e : V≃W) (he : ∀ a b,e a<e b ↔ a<b) (w : W→W→α) (a b : V) :
seed (fun i j=>w (e i) (e j)) a b=seed w (e a) (e b) := by
simp only [seed,edge,e.injective.eq_iff,he]
lemma advance_reindex (e : V≃W) (w : W→W→α) (a b : V) :
advance (fun i j=>w (e i) (e j)) a b=advance w (e a) (e b) := by
unfold advance
simp only [e.injective.eq_iff]
split_ifs
· rfl
· congr 1
apply le_antisymm
· apply Finset.sup'_le
intro c _
exact Finset.le_sup' (fun k=>min (w (e a) k) (w k (e b))) (Finset.mem_univ (e c))
· apply Finset.sup'_le
intro c _
obtain ⟨d,rfl⟩:=e.surjective c
exact Finset.le_sup' (fun k=>min (w (e a) (e k)) (w (e k) (e b))) (Finset.mem_univ d)
lemma close_reindex (e : V≃W) (he : ∀ a b,e a<e b ↔ a<b) (w : W→W→α) (k : ℕ) :
close (fun i j=>w (e i) (e j)) k=fun i j=>close w k (e i) (e j) := by
induction k with
| zero=>funext a b; exact seed_reindex e he w a b
| succ k ih=>
simp only [close,ih]
funext a b
exact advance_reindex e _ a b
end MatchingFPRAS.BottleneckAlgorithm
end OAI
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