IsBoundary₁ — Mathlib

English

If x ∈ cycles₂(Rep.ofDistribMulAction k G A), then IsCycle₂ x.

Русский

Если x ∈ cycles₂(Rep.ofDistribMulAction k G A), то x является IsCycle₂.

LaTeX

$$$\text{mem\_cycles₂\_iff}(A := Rep.ofDistribMulAction k G A)\; x \Rightarrow IsCycle_2 x$$$

Lean4

/-- A finsupp `x : G →₀ A` satisfies the 1-boundary condition if there's a finsupp
`∑ aᵢ·(gᵢ, hᵢ) : G × G →₀ A` such that `∑ (gᵢ⁻¹ • aᵢ)·hᵢ - aᵢ·gᵢhᵢ + aᵢ·gᵢ = x`. -/
def IsBoundary₁ (x : G →₀ A) : Prop :=
  ∃ y : G × G →₀ A, y.sum (fun g a => single g.2 (g.1⁻¹ • a) - single (g.1 * g.2) a + single g.1 a) = x

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