English
If f and g are disjoint, then the cycle factors of f·g equal the union of cycle factors of f and g.
Русский
Если f и g дисjoint, то cycleFactorsFinset(f·g) равно объединению cycleFactorsFinset(f) и cycleFactorsFinset(g).
LaTeX
$$cycleFactorsFinset (f * g) = cycleFactorsFinset f ∪ cycleFactorsFinset g$$
Lean4
theorem cycleFactorsFinset_mul_eq_union {f g : Perm α} (h : Disjoint f g) :
cycleFactorsFinset (f * g) = cycleFactorsFinset f ∪ cycleFactorsFinset g :=
by
rw [cycleFactorsFinset_eq_finset]
refine ⟨?_, ?_, ?_⟩
· simp [or_imp, mem_cycleFactorsFinset_iff, forall_swap]
· rw [coe_union, Set.pairwise_union_of_symmetric Disjoint.symmetric]
exact
⟨cycleFactorsFinset_pairwise_disjoint _, cycleFactorsFinset_pairwise_disjoint _, fun x hx y hy _ =>
h.mono (mem_cycleFactorsFinset_support_le hx) (mem_cycleFactorsFinset_support_le hy)⟩
· rw [noncommProd_union_of_disjoint h.disjoint_cycleFactorsFinset]
rw [cycleFactorsFinset_noncommProd, cycleFactorsFinset_noncommProd]
