Theorem 11  Given a CSP and a variable ordering, MCk visits all the nodes that MCk vistis.

## Proof

4.3 Relationship Between BJ k and MC k k

$$\mathbf{\Sigma}^{k}$$

k

$$\leq\ k\leq\ n$$

Lemma 12  If CBJ performs a one-level backjump from a deeper variable x i to a shallower variable x h, the node th at the level of xh is one-inconsistent.

Proof Si X i x x h

x P r si ti x t ; P C vars C t i x P t n Si vars C  C Si C x x t x x X X i P t n Pr si P t n s i N P t n t h X h 
Lemma 13 If CBJ performs a k-level backjump from a deeper variable xi to a shallower variable x h , the current node t h at the level of x h is k-inconsistent.

Proof Si h l h

$$\begin{array}{c c c}{{}}&{{x_{i}}}&{{}}&{{x_{h}}}\\ {{}}&{{x_{i}}}&{{x_{h}}}&{{}}\\ {{}}&{{t_{h}}}&{{k}}\end{array}$$
$$k\,-\,$$
$$\begin{array}{ccccccccc}&&k&-&&&&&&&&&\\ &&&&&P&_{t_h}&S_i\\ &&&&&a&&&&&t_i\\ &&&&&t_h&&&&&\\ &&&&&t_h&&&&&\\ &&&&&\leq&l&&k&&&&&\\ &&&&&x_j&&&&&&&&&\\ &&&&&&&&&&a&&&&&\\ &&&&&P&_{t_h}&S_i&&x_i&&a&&\\ &&t_i&S_j&\subseteq&t_h&S_i&&x_i&&a&&\\ &&k-&&&&&&&&&t_h&&&&&\\ &&&&&k&&&&&t_h&&&&&\\ &&&&&t_h&&&&&\end{array}$$

k k

$\begin{array}{ccccc}&&k&&\\ &x_{i}&&k&&P\\ &x_{i}&&x_{i}&&a\\ &l&&&&&x_{j}\\ &&k&&&&&S_{j}\\ &&P\ _{t_{i}}\ S_{j}&&&&&\\ &&k\ -&&&&\\ &&P\ _{t_{i}}\ S_{j}&&&&&\\ &&P\ _{t_{h}}\ S_{i}&&&&&\\ &&x_{h}&&k&&&&&\end{array}$
$$\begin{array}{l l l}{{k}}&{}&{{x_{i}}}\\ {}&{}&{{k}}\\ {}&{}&{{k}}\\ {{x_{i}}}&{}&{{a}}\end{array}$$