English
The GFP of GFP under a compositional operator equals the GFP of the operator applied to GFP on the diagonal.
Русский
Фиксированная точка GFP после применения GFP к композиции равна GFP точке, полученной от оператора на диагонали.
LaTeX
$$$\mathrm{gfp}(\mathrm{gfp}(f) \circ h) = \mathrm{gfp}(f) \circ h^{\mathrm{onDiag}}$.$$
Lean4
theorem lfp_lfp (h : α →o α →o α) : (lfp.comp h).lfp = h.onDiag.lfp :=
by
let a := (lfp.comp h).lfp
refine (lfp_le _ ?_).antisymm (lfp_le _ (Eq.le ?_))
· exact lfp_le _ h.onDiag.map_lfp.le
have ha : (lfp ∘ h) a = a := (lfp.comp h).map_lfp
calc
h a a = h a (h a).lfp := congr_arg (h a) ha.symm
_ = (h a).lfp := (h a).map_lfp
_ = a := ha
