cycleOf_eq — Mathlib

English

If f is a cycle and x is not fixed by f, then the cycleOf f x equals f.

Русский

Если f — цикл и x не зафиксирована, то cycleOf f x равен f.

LaTeX

$$cycleOf_eq [DecidableRel f.SameCycle] (hf : IsCycle f) (hx : f x ≠ x) : cycleOf f x = f$$

Lean4

theorem cycleOf_eq [DecidableRel f.SameCycle] (h : SameCycle f x y) : cycleOf f x = cycleOf f y :=
  by
  ext z
  rw [Equiv.Perm.cycleOf_apply]
  split_ifs with hz
  · exact (h.symm.trans hz).cycleOf_apply.symm
  · exact (cycleOf_apply_of_not_sameCycle (mt h.trans hz)).symm

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