  We prove Lusztig's conjectures P1-P15 for Coxeter groups with complete graph,
using deceasing induction on $ \mathbf{a} $-values and a kind of decomposition
formula of Kazhdan-Lusztig basis elements. As a byproduct, we give a
description of the left, right, and two-sided cells. In the appendix, we prove
P1-P15 for right-angled Coxeter groups by the same methods.
