| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators Classical |
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| namespace BinaryCoordinateSweeps |
| variable {I : Type*} [DecidableEq I] |
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| def alternatingSubsetSum (U : Finset I) (s : ℝ) (p : Finset I → ℝ) : ℝ := |
| ∑A ∈ U.powerset, (-1:ℝ)^(U.card-A.card)*s^A.card*p A |
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| lemma alternatingSubsetSum_of_isolated (U : Finset I) {s : ℝ} (hs : s≠0) |
| (p : Finset I → ℝ) (i : I) (hi : i∈U) |
| (hp : ∀A ⊆ U.erase i, p (insert i A)=s⁻¹*p A) : |
| alternatingSubsetSum U s p=0 := by |
| have hUi : U=insert i (U.erase i) := (Finset.insert_erase hi).symm |
| have hn : i∉U.erase i := Finset.notMem_erase i U |
| unfold alternatingSubsetSum |
| rw [hUi,Finset.sum_powerset_insert hn,← Finset.sum_add_distrib] |
| apply Finset.sum_eq_zero |
| intro A hA |
| have hAU : A⊆U.erase i := Finset.mem_powerset.mp hA |
| have hiA : i∉A := fun h => hn (hAU h) |
| have hcard := Finset.card_le_card hAU |
| rw [Finset.card_insert_of_notMem hiA,hp A hAU] |
| have he : (insert i (U.erase i)).card-A.card = |
| ((insert i (U.erase i)).card-(A.card+1))+1 := by |
| rw [Finset.card_insert_of_notMem hn] |
| omega |
| rw [he,pow_succ,pow_succ] |
| simp only [mul_assoc] |
| field_simp |
| ring |
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| end BinaryCoordinateSweeps |
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| end |
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| end OAI |
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