| import OAI.Combinatorics.MatchingPSD.CutGeometry |
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| namespace OAI |
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| universe u_1 u_2 u_3 u_4 u_5 u_6 |
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| 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] |
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| 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) |
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| 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) |
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| 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] |
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| 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] |
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| 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 |
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| 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] |
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| end CellAverages |
| end MatchingSmoothing |
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| end OAI |
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