File size: 3,333 Bytes
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 | import OAI.Combinatorics.MatchingPSD.CutGeometry
namespace OAI
universe u_1 u_2 u_3 u_4 u_5 u_6
namespace MatchingSmoothing
open scoped BigOperators
attribute [local instance] Classical.propDecidable
section CellAverages
variable {A : Type u_1} {B : Type u_2} {C : Type u_3} {I : Type u_4} [Fintype A] [Fintype B] [Fintype C] [Fintype I] [DecidableEq I]
noncomputable def extensionAverage (i : I) (f : Pairing ((A ⊕ B) ⊕ (I × C)) → ℝ)
(M : Pairing ((A ⊕ B) ⊕ C)) : ℝ :=
𝔼 N : {j : I // j ≠ i} → Pairing C, f (extendLocal i M N)
noncomputable def cellAverage (i : I) (f : Pairing ((A ⊕ B) ⊕ (I × C)) → ℝ)
(x : Fin 3 → ℝ) : ℝ :=
predicateAverage (fun M => crossVector threeColor M = x) (extensionAverage i f)
omit [Fintype A] [Fintype B] in
lemma extensionAverage_relabel (i : I) (p : Equiv.Perm ((A ⊕ B) ⊕ C))
(f : Pairing ((A ⊕ B) ⊕ (I × C)) → ℝ) (M : Pairing ((A ⊕ B) ⊕ C)) :
extensionAverage i f (Pairing.relabel p M) =
extensionAverage i (fun N => f (Pairing.relabel (liftLocal i p) N)) M := by
simp only [extensionAverage, extendLocal_relabel]
lemma expect_pair_product {X : Type u_5} {Y : Type u_6} [Fintype X] [Fintype Y] (f : X × Y → ℝ) :
(𝔼 p : X × Y, f p) = 𝔼 x : X, 𝔼 y : Y, f (x,y) := by
rw [← Finset.univ_product_univ, Finset.expect_product]
lemma cellAverage_star (i : I) (f : Pairing ((A ⊕ B) ⊕ (I × C)) → ℝ) (x : ℝ) :
cellAverage i f ![x,0,0] =
𝔼 M : {M : Pairing (A ⊕ B) // tableCoordinate abColor (0,1) M = x},
𝔼 N : I → Pairing C, f (Pairing.sum M.1 (Pairing.cellwise N)) := by
classical
rw [cellAverage, predicateAverage_eq_expect]
have he := Fintype.expect_equiv (starFiberEquiv (A := A) (B := B) (C := C) x)
(fun p => extensionAverage i f (Pairing.sum p.1.1 p.2))
(fun p => extensionAverage i f p.1) (fun _ => rfl)
calc
_ = 𝔼 p : {M : Pairing (A ⊕ B) // tableCoordinate abColor (0,1) M = x} × Pairing C,
extensionAverage i f (Pairing.sum p.1.1 p.2) := by
convert he.symm using 1
congr 1
ext; simp
_ = _ := ?_
rw [expect_pair_product]
apply Finset.expect_congr rfl
intro M _
simp only [extensionAverage]
calc
_ = 𝔼 p : Pairing C × ({j : I // j ≠ i} → Pairing C),
f (extendLocal i (Pairing.sum M.1 p.1) p.2) := by
convert (expect_pair_product (fun p : Pairing C × ({j : I // j ≠ i} → Pairing C) =>
f (extendLocal i (Pairing.sum M.1 p.1) p.2))).symm using 1
_ = _ := (Fintype.expect_equiv (Equiv.piSplitAt i (fun _ : I => Pairing C))
(fun N => f (Pairing.sum M.1 (Pairing.cellwise N)))
(fun p => f (extendLocal i (Pairing.sum M.1 p.1) p.2))
(fun N => by rw [Equiv.piSplitAt_apply, extendLocal_sum])).symm
lemma cellAverage_const (i : I) (f : Pairing ((A ⊕ B) ⊕ (I × C)) → ℝ)
(x : Fin 3 → ℝ) (c : ℝ) (hM : ∃ M : Pairing ((A ⊕ B) ⊕ C), crossVector threeColor M = x)
[Nonempty (Pairing C)]
(hf : ∀ M, crossVector threeColor M = x → ∀ N, f (extendLocal i M N) = c) :
cellAverage i f x = c := by
apply predicateAverage_of_constant _ _ hM
intro M hM
simp only [extensionAverage, hf M hM, Finset.expect_const, Finset.univ_nonempty]
end CellAverages
end MatchingSmoothing
end OAI
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