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9b6e014 | 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 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps
variable {I : Type*} [DecidableEq I]
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
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
end BinaryCoordinateSweeps
end
end OAI
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