English
Given a bijection f: α → Subtype p and two disjoint permutations σ, τ on α, the permutations obtained by extending σ and τ to the larger domain β via f remain disjoint on β.
Русский
Дано биекция f:α → Subtype p и две перестановки σ, τ на α, не связанные, тогда их продолжения на большем домене β по f остаются дисjoint на β.
LaTeX
$$$\\text{If } f: α \\simeq \\mathrm{Subtype}(p) \\text{ and } \\sigma,\\tau \\in \\mathrm{Perm}(α) \\text{ with } \\mathrm{Disjoint}(\\sigma,\\tau),\\ \\text{then } \\mathrm{Disjoint}(\\sigma.{\\extendDomain}f, \\tau.{\\extendDomain}f).$$$
Lean4
theorem extendDomain {p : β → Prop} [DecidablePred p] (f : α ≃ Subtype p) {σ τ : Perm α} (h : Disjoint σ τ) :
Disjoint (σ.extendDomain f) (τ.extendDomain f) := by
intro b
by_cases pb : p b
·
refine (h (f.symm ⟨b, pb⟩)).imp ?_ ?_ <;>
· intro h
rw [extendDomain_apply_subtype _ _ pb, h, apply_symm_apply, Subtype.coe_mk]
· left
rw [extendDomain_apply_not_subtype _ _ pb]
