Appendix A: Proof of (15 ) In this appendix, we prove Proposition 1 . From the definition, \(\mathrm{up}[B\otimes C]\) is expressed by
$$\begin{aligned} \mathrm{up}[B \otimes C] = \mathrm{up}\begin{bmatrix} {\displaystyle \max _{1\le k\le M}}(b_{1k}+c_{k1})&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{1k}+c_{kN}) \\ \vdots&\ddots&\vdots \\ {\displaystyle \max _{1\le k\le M}}(b_{Nk}+c_{k1})&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{Nk}+c_{kN}) \end{bmatrix}. \end{aligned}$$
(44)
Applying (12 ) and (13 ) to the first column, (44 ) is expanded as the maximum of M UPs as below.
$$\begin{aligned}&\mathrm{up}[B\otimes C]\nonumber \\&\quad =\max \left( \mathrm{up}\begin{bmatrix} b_{11}+c_{11}&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{1k}+c_{kN}) \\ \vdots&\ddots&\vdots \\ b_{N1}+c_{11}&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{Nk}+c_{kN}) \end{bmatrix}, \right. \nonumber \\&\qquad \mathrm{up}\begin{bmatrix} b_{12}+c_{21}&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{1k}+c_{kN}) \\ \vdots&\ddots&\vdots \\ b_{N2}+c_{21}&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{Nk}+c_{kN}) \end{bmatrix}, \ \dots , \nonumber \\&\qquad \left. \mathrm{up}\begin{bmatrix} b_{1M}+c_{M1}&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{1k}+c_{kN}) \\ \vdots&\ddots&\vdots \\ b_{NM}+c_{M1}&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{Nk}+c_{kN}) \end{bmatrix}\right) \nonumber \\&\quad = \max _{1\le k_1\le M} \left( c_{k_11}+\mathrm{up}\begin{bmatrix} b_{1k_1}&\ldots&\max _{1\le k\le M}(b_{1k}+c_{kN}) \\ \vdots&\ddots&\vdots \\ b_{Nk_1}&\ldots&\max _{1\le k\le M}(b_{Nk}+c_{kN}) \end{bmatrix}\right) . \end{aligned}$$
(45)
Applying similar procedure to the other columns, we obtain
$$\begin{aligned} \mathrm{up}[B\otimes C]= \max _{1\le k_1, k_2, \dots , k_N\le M} \left( \sum _{1\le i\le N} c_{k_ii} +\mathrm{up}\begin{bmatrix} b_{1k_1}&\ldots&b_{1k_N} \\ \vdots&\ddots&\vdots \\ b_{Nk_1}&\ldots&b_{Nk_N} \end{bmatrix}\right) . \end{aligned}$$
(46)
It is equivalent to
$$\begin{aligned} \mathrm{up}[B\otimes C]=\max _{1\le j_1\le \dots \le j_N\le M} \left( \max _{\pi '}\left( \sum _{1\le i\le N} c_{\pi ' _ii} +\mathrm{up}\begin{bmatrix} b_{1\pi '_1}&\ldots&b_{1\pi '_N} \\ \vdots&\ddots&\vdots \\ b_{N\pi '_1}&\ldots&b_{N\pi '_N} \end{bmatrix}\right) \right) , \end{aligned}$$
(47)
where \(\pi '=(\pi '_1, \pi '_2, \dots , \pi '_N)\) is a set of all possible permutations of \(\{j_1, j_2, \dots , j_N \}\) . In particular, the UP of the matrix whose columns are exchanged is the same as original one from (11 ). Therefore, we obtain
$$\begin{aligned} \mathrm{up}[B\otimes C]= & {} \max _{1\le j_1\le \dots \le j_N\le M} \left( \max _{\pi ' } \sum _{1\le i\le N} c_{\pi '_ii} +\mathrm{up}\begin{bmatrix} b_{1j_1}&\ldots&b_{1j_N} \\ \vdots&\ddots&\vdots \\ b_{Nj_1}&\ldots&b_{Nj_N} \end{bmatrix}\right) \nonumber \\= & {} \max _{1\le j_1\le \dots \le j_N\le M} \left( \mathrm{up}[C]^{j_1\dots j_N}_{1\dots N} +\mathrm{up}[B]^{1\dots N}_{j_1\dots j_N} \right) . \end{aligned}$$
(48)
Thus (15 ) holds.
Appendix B: Proof of (16 ) In this appendix, we express
$$\begin{aligned} \varvec{a} = \begin{bmatrix} a_1\\ a_2 \\ \vdots \\ a_N\end{bmatrix}, \quad \varvec{b} = \begin{bmatrix} b_1\\ b_2 \\ \vdots \\ b_N\end{bmatrix}, \quad \varvec{c} = \begin{bmatrix} c_1\\ c_2 \\ \vdots \\ c_N\end{bmatrix}, \end{aligned}$$
respectively. Then \(\mathrm{up}[D_{l , m, n}]+\mathrm{up}[D_{l' , m', n'}]\) is expressed by
$$\begin{aligned}&\mathrm{up}[D_{l , m, n}] +\mathrm{up}[D_{l' , m' , n' }] \nonumber \\&\quad = \max _{\pi , \pi ' }\Bigl ( a_{\pi _1}+\dots + a_{\pi _l}+b_{\pi _{l+1}}+\dots +b_{\pi _{l+m}}+ c_{\pi _{l+m+1}}+\dots + c_{\pi _N} \nonumber \\&\qquad +\, a_{\pi '_1}+\dots + a_{\pi ' _{l' }}+b_{\pi ' _{l' +1}}+\dots +b_{\pi ' _{l' +m'}}+ c_{\pi '_{l' +m' +1}}+\dots + c_{\pi ' _N}\Bigr ), \end{aligned}$$
(49)
namely, \(\mathrm{up}[D_{l , m, n}]+\mathrm{up}[D_{l' , m', n'}]\) can be expressed by
$$\begin{aligned}&a_{\pi _1}+\dots + a_{\pi _l}+b_{\pi _{l+1}}+\dots +b_{\pi _{l+m}}+ c_{\pi _{l+m+1}}+\dots + c_{\pi _N} \nonumber \\&\quad +\, a_{\pi '_1}+\dots + a_{\pi ' _{l' }}+b_{\pi ' _{l' +1}}+\dots +b_{\pi ' _{l' +m'}}+ c_{\pi '_{l' +m' +1}}+\dots + c_{\pi ' _N} \end{aligned}$$
(50)
for certain permutations \(\pi \) and \(\pi '\) . In particular, due to \(m>m'\) , there exists j such that
$$\begin{aligned}&j\in \{ \pi _{l+1 }, \pi _{l +2 }, \dots , \pi _{l+m }\}\quad \text {and }\\&\quad j\in \{ \pi '_1 , \pi '_2 , \dots , \pi '_{l'}\} \cup \{ \pi '_{l'+m'+1} , \pi '_{l'+m'+2} , \dots , \pi '_N\}. \end{aligned}$$
First, let us consider in the case there exists \(j_0\) such that
$$\begin{aligned} j_0\in \{ \pi _{l+1 }, \pi _{l +2 }, \dots , \pi _{l+m }\}\quad \text {and } \quad j_0\in \{ \pi '_1 , \pi '_2 , \dots , \pi '_{l'}\}. \end{aligned}$$
(51)
Then \(\mathrm{up}[D_{l, m, n}] +\mathrm{up}[D_{l' , m' , n' }]\) can be expanded as
$$\begin{aligned} \mathrm{up}[D_{l, m, n}] +\mathrm{up}[D_{l' , m' , n' }] =\mathrm{up}\left[ D_{l, m, n} \begin{bmatrix} j_0 \\ l+1 \end{bmatrix}\right] +b_{j_0} + \mathrm{up}\left[ D_{l' , m' , n'} \begin{bmatrix} j_0 \\ 1 \end{bmatrix}\right] +a_{j_0} \end{aligned}$$
(52)
where \(D \begin{bmatrix} j \\ k \end{bmatrix}\) denotes the \((N-1)\times (N-1)\) matrix obtained by eliminating the j th row and the k th column from D . On the other hand, \(\mathrm{up}[D_{l+1 , m-1 , n}] +\mathrm{up}[D_{l'-1 , m'+1 , n' }]\) can be evaluated as
$$\begin{aligned}&\mathrm{up}[D_{l+1 , m-1 , n}] +\mathrm{up}[D_{l'-1 , m'+1 , n' }] \nonumber \\&\quad \ge \mathrm{up}\left[ D_{l+1 , m-1 , n}\begin{bmatrix} j_0 \\ 1 \end{bmatrix}\right] +a_{j_0} +\mathrm{up}\left[ D_{l'-1 , m'+1 , n' }\begin{bmatrix} j_0 \\ l' \end{bmatrix}\right] +b_{j_0}. \end{aligned}$$
(53)
From (52 ) and (53 ), we obtain
$$\begin{aligned} \mathrm{up}[D_{l+1 , m-1 , n}] +\mathrm{up}[D_{l'-1 , m'+1 , n' }]\ge \mathrm{up}[D_{l , m , n}] +\mathrm{up}[D_{l' , m' , n' }] \end{aligned}$$
(54)
since
$$\begin{aligned} \mathrm{up}\left[ D_{l, m, n} \begin{bmatrix} j_0 \\ l+1 \end{bmatrix}\right]= & {} \mathrm{up}\left[ D_{l+1 , m-1 , n}\begin{bmatrix} j_0 \\ 1 \end{bmatrix}\right] ,\nonumber \\ \mathrm{up}\left[ D_{l' , m' , n'} \begin{bmatrix} j_0 \\ 1 \end{bmatrix}\right]= & {} \mathrm{up}\left[ D_{l'-1 , m'+1 , n' }\begin{bmatrix} j_0 \\ l' \end{bmatrix}\right] \end{aligned}$$
(55)
hold.
Next we consider the case there is no \(j_0\) such that (51 ). Then there exists \(j_1\) such that
$$\begin{aligned} j_1\in \{ \pi _{l+1 }, \pi _{l +2 }, \dots , \pi _{l+m }\}\quad \text {and } \quad j_1\in \{ \pi '_{l'+m'+1} , \pi '_{l'+m'+2} , \dots , \pi '_N\} . \end{aligned}$$
In addition, due to \(l<l'\) , there also exists \(j_2\) such that
$$\begin{aligned} j_2\in \{ \pi _{l +m+1} , \pi _{l+m+2} , \dots , \pi _N \} \quad \text {and } \quad j_2\in \{ \pi '_1 , \pi '_2 , \dots , \pi '_{l'} \}. \end{aligned}$$
Thus, \(\mathrm{up}[D_{l, m, n}] +\mathrm{up}[D_{l' , m' , n' }]\) can be expanded as
$$\begin{aligned}&\mathrm{up}[D_{l, m, n}] +\mathrm{up}[D_{l' , m' , n' }] \nonumber \\&\quad = \mathrm{up}\left[ D_{l, m , n} \begin{bmatrix} j_1&j_2 \\ l+1&l +m+1\end{bmatrix}\right] +b_{j_1}+c_{j_2} \nonumber \\&\qquad +\, \mathrm{up}\left[ D_{l' , m' , n'} \begin{bmatrix}j_1&j_2 \\ l'+m'+1&1 \end{bmatrix}\right] +c_{j_1}+a_{j_2} . \end{aligned}$$
(56)
On the other hand \(\mathrm{up}[D_{l+1 , m-1 , n}] +\mathrm{up}[D_{l'-1 , m'+1 , n' }]\) can be evaluated as
$$\begin{aligned}&\mathrm{up}[D_{l+1 , m-1 , n}] +\mathrm{up}[D_{l'-1 , m'+1 , n' }] \nonumber \\&\quad \ge \mathrm{up}\left[ D_{l+1 , m-1 , n }\begin{bmatrix} j_1&j_2 \\ l+m+1&1 \end{bmatrix}\right] +c_{j_1}+a_{j_2} \nonumber \\&\qquad +\,\mathrm{up}\left[ D_{l'-1 , m'+1 , n' }\begin{bmatrix} j_1&j_2 \\ l'&l'+m'+1 \end{bmatrix}\right] +b_{j_1}+c_{j_2}. \end{aligned}$$
(57)
Thus we obtain \(\mathrm{up}[D_{l+1 , m-1 , n}] +\mathrm{up}[D_{l'-1 , m'+1 , n' }]\ge \mathrm{up}[D_{l , m , n}] +\mathrm{up}[D_{l' , m' , n' }]\) since
$$\begin{aligned} \mathrm{up}\left[ D_{l, m , n} \begin{bmatrix} j_1&j_2 \\ l+1&l +m+1\end{bmatrix}\right]= & {} \mathrm{up}\left[ D_{l+1 , m-1 , n }\begin{bmatrix} j_1&j_2 \\ l+m+1&1 \end{bmatrix}\right] , \nonumber \\ \mathrm{up}\left[ D_{l' , m' , n'} \begin{bmatrix}j_1&j_2 \\ l'+m'+1&1 \end{bmatrix}\right]= & {} \mathrm{up}\left[ D_{l'-1 , m'+1 , n' }\begin{bmatrix} j_1&j_2 \\ l'&l'+m'+1 \end{bmatrix}\right] \end{aligned}$$
(58)
hold.
Appendix C: Proof of (35 ) We prove (35 ) in the case of \(M=1, 2, 3\) and any integer N . We can assume
$$\begin{aligned} \varphi _i(s) = \max _{1\le j\le M}( c_{ij}+p_js) \end{aligned}$$
(59)
and
$$\begin{aligned} p_1\le p_2\le \dots \le p_M \end{aligned}$$
(60)
without a loss of generality. One can prove (35 ) in the case of \(M=1\) . In the case of \(M=2\) , we add \(-(c_{i1}+c_{i2})/2\) to each i th row and \(-(p_1s+p_2s)/2\) to each column \(\varvec{\varphi }(s)\) for all the UP in (35 ). In other words, for the property (12 ), we add \(-\sum _{i=1}^N(c_{i1}+c_{i2})- \sum _{i=1}^N (p_1+p_2)i- (p_1+p_2)(N+1-k_1-k_2-k_3)/2\) to both sides in (35 ). Then each \(\varphi _i(s)\) is reduced as
$$\begin{aligned} \varphi _i(s)-\frac{c_{i1}+c_{i2}+p_1s+p_2s}{2} = \frac{1}{2}|c_{i1}-c_{i2}+(p_1-p_2)s|. \end{aligned}$$
Thus, (35 ) can be rewritten as
$$\begin{aligned}&\mathrm{up}\left[ \varvec{\varphi }(0) \cdots \widehat{\varvec{\varphi }(k_2)}\cdots \varvec{\varphi }(N)\right] + \mathrm{up}\left[ \varvec{\varphi }(0)\cdots \widehat{\varvec{\varphi }(k_1)}\cdots \widehat{\varvec{\varphi }(k_3)}\cdots \varvec{\varphi }(N+1)\right] \nonumber \\&\quad \ge \mathrm{up}\left[ \varvec{\varphi }(0) \cdots \widehat{\varvec{\varphi }(k_1)}\cdots \varvec{\varphi }(N)\right] + \mathrm{up}\left[ \varvec{\varphi }(0)\cdots \widehat{\varvec{\varphi }(k_2)}\cdots \widehat{\varvec{\varphi }(k_3)}\cdots \varvec{\varphi }(N+1)\right] ,\nonumber \\ \end{aligned}$$
(61)
where
$$\begin{aligned} {\varvec{\varphi }(s)}= \begin{pmatrix} |c'_{1}+p s| \\ |c'_{2}+p s| \\ \dots \\ |c'_{N}+p s| \end{pmatrix} \end{aligned}$$
(62)
with arbitrary parameters \(c'_i\) and \(p'\) . This inequality has been proved in [6 ]. Hence (35 ) in the case of \(M=2\) holds.
We consider in the case of \(M=3\) . Using (15 ), \(\mathrm{up}[\varvec{\varphi }(0) \cdots \widehat{\varvec{\varphi }(k_2)}\cdots \varvec{\varphi }(N)]\) can be expanded as
$$\begin{aligned}&\mathrm{up}\left[ \varvec{\varphi }(0) \cdots \widehat{\varvec{\varphi }(k_2)}\cdots \varvec{\varphi }(N)\right] \\&\quad = \mathrm{up}\begin{bmatrix} \begin{bmatrix} c_{11}&c_{12}&c_{13} \\ c_{21}&c_{22}&c_{23} \\ \vdots&\vdots&\vdots \\ c_{N1}&c_{N2}&c_{N3} \end{bmatrix} \otimes \begin{bmatrix} 0&p_1&2p_1&\ldots&\widehat{k_2p_1}&\ldots&Np_1\\ 0&p_2&2p_2&\ldots&\widehat{k_2p_2}&\ldots&Np_2\\ 0&p_3&2p_3&\ldots&\widehat{k_2p_3}&\ldots&Np_3 \end{bmatrix} \end{bmatrix}\\&\quad = \max _{1\le i_1\le \dots \le i_N\le 3} \left( \mathrm{up}\begin{bmatrix} c_{1i_1}&c_{1i_2}&\ldots&c_{1i_N} \\ c_{2i_1}&c_{2i_2}&\ldots&c_{2i_N} \\ \vdots&\vdots&\ddots&\vdots \\ c_{Ni_1}&c_{Ni_2}&\ldots&c_{Ni_N} \end{bmatrix} + \mathrm{up}\begin{bmatrix} 0&p_{i_1}&2p_{i_1}&\ldots&\widehat{k_2p_{i_1}}&\ldots&Np_{i_1}\\ 0&p_{i_2}&2p_{i_2}&\ldots&\widehat{k_2p_{i_2}}&\ldots&Np_{i_2}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots \\ 0&p_{i_N}&2p_{i_N}&\ldots&\widehat{k_2p_{i_N}}&\ldots&Np_{i_N} \end{bmatrix} \right) \\&\quad = \max _{1\le i_1\le \dots \le i_N\le 3}\left( \mathrm{up}[C_{i_1i_2\dots i_N}]+ \sum _{l=1}^{k_2} (l-1) p_{i_l}+\sum _{l=k_2+1}^{N} l p_{i_l}\right) , \end{aligned}$$
where \( C_{i_1i_2\dots i_N}\) denotes
$$\begin{aligned} C_{i_1i_2\dots i_N}= \begin{bmatrix} c_{1i_1}&c_{1i_2}&\ldots&c_{1i_N} \\ c_{2i_1}&c_{2i_2}&\ldots&c_{2i_N} \\ \vdots&\vdots&\ddots&\vdots \\ c_{Ni_1}&c_{Ni_2}&\ldots&c_{Ni_N} \end{bmatrix}. \end{aligned}$$
By similar procedure, we obtain
$$\begin{aligned}&\mathrm{up}[\varvec{\varphi }(0) \cdots \widehat{\varvec{\varphi }(k_2)}\cdots \varvec{\varphi }(N)]+ \mathrm{up}[\varvec{\varphi }(0)\cdots \widehat{\varvec{\varphi }(k_1)}\cdots \widehat{\varvec{\varphi }(k_3)}\cdots \varvec{\varphi }(N+1)]\nonumber \\&\quad = \max _{I, J}\left( \mathrm{up}[C_{i_1i_2\dots i_N}]+\mathrm{up}[C_{j_1j_2\dots j_N}]+\sum _{l=1}^N l p_{i_l}\right. \nonumber \\&\qquad \left. + \sum _{l=1}^N lp_{j_l}-\sum _{l=1}^{k_2} p_{i_l}-\sum _{l=1}^{k_1} p_{j_l} +\sum _{l=k_3}^N p_{j_l}\right) ,\end{aligned}$$
(63)
$$\begin{aligned}&\mathrm{up}[\varvec{\varphi }(0) \cdots \widehat{\varvec{\varphi }(k_1)}\cdots \varvec{\varphi }(N)]+ \mathrm{up}[\varvec{\varphi }(0)\cdots \widehat{\varvec{\varphi }(k_2)}\cdots \widehat{\varvec{\varphi }(k_3)}\cdots \varvec{\varphi }(N+1)]\nonumber \\&\quad =\max _{I', J'}\left( \mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] +\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] +\sum _{l=1}^N l p_{i'_l}\right. \nonumber \\&\qquad \left. + \sum _{l=1}^N lp_{j'_l}-\sum _{l=1}^{k_1} p_{i'_l}-\sum _{l=1}^{k_2} p_{j'_l} +\sum _{l=k_3}^N p_{j'_l}\right) . \end{aligned}$$
(64)
Here we denote \(\max _{1\le i_1\le \dots \le i_N\le 3, 1\le j_1\le \dots \le j_N\le 3}\) as \(\max _{I, J}\) and \(\sum _{l=m}^n\) is defined as 0 when \(m>n\) . Let us consider the argument of RHS of (64 ):
$$\begin{aligned} \mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] +\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] + \sum _{l=1}^N l p_{i'_l}+ \sum _{l=1}^N lp_{j'_l}-\sum _{l=1}^{k_1} p_{i'_l}-\sum _{l=1}^{k_2} p_{j'_l} +\sum _{l=k_3}^N p_{j'_l}. \end{aligned}$$
(65)
Equation (35 ) in the case of \(M=3\) holds if we show (65 ) is less than or equal to RHS of (63 ) for any \(i'_1, i'_2, \dots i'_N\) , \(j'_1, j'_2, \dots , j'_N\) . Hereafter we fix \(i'_1, i'_2, \dots i'_N\) , \(j'_1, j'_2, \dots , j'_N\) . First we compare (65 ) and the arguments of RHS of (63 ) associated with \(i_l=i'_l\) , \(j_i=j'_l\) or \(i_l=j'_l\) , \(j_i=i'_l\) . Then, we obtain
$$\begin{aligned}&\max _{I, J}\left( \mathrm{up}[C_{i_1i_2\dots i_N}]+\mathrm{up}[C_{j_1j_2\dots j_N}]+\sum _{l=1}^N l p_{i_l}+ \sum _{l=1}^N lp_{j_l}-\sum _{l=1}^{k_2} p_{i_l}-\sum _{l=1}^{k_1} p_{j_l} +\sum _{l=k_3}^N p_{j_l}\right) \nonumber \\&\qquad - \left( \mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] +\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] + \sum _{l=1}^N l p_{i'_l}+ \sum _{l=1}^N lp_{j'_l}-\sum _{l=1}^{k_1} p_{i'_l}-\sum _{l=1}^{k_2} p_{j'_l} +\sum _{l=k_3}^N p_{j'_l}\right) \nonumber \\&\quad \ge \max \left( \mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] +\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] + \sum _{l=1}^N l p_{i'_l}+ \sum _{l=1}^N lp_{j'_l}-\sum _{l=1}^{k_2} p_{i'_l}-\sum _{l=1}^{k_1} p_{j'_l} +\sum _{l=k_3}^N p_{j'_l},\right. \nonumber \\&\mathrm{up}\left. \left[ C_{j'_1j'_2\dots j'_N}\right] +\mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] + \sum _{l=1}^N l p_{j'_l}+ \sum _{l=1}^N lp_{i'_l}-\sum _{l=1}^{k_2} p_{j'_l}-\sum _{l=1}^{k_1} p_{i'_l} +\sum _{l=k_3}^N p_{i'_l}\right) \nonumber \\&\qquad -\left( \mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] +\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] + \sum _{l=1}^N l p_{i'_l}+ \sum _{l=1}^N lp_{j'_l}-\sum _{l=1}^{k_1} p_{i'_l}-\sum _{l=1}^{k_2} p_{j'_l} +\sum _{l=k_3}^N p_{j'_l}\right) \nonumber \\&\quad = \max \left( \sum _{l=k_1+1}^{k_2} (-p_{i'_l}+p_{j'_l}) , \quad \sum _{l=k_3}^N (p_{i'_l} - p_{j'_l})\right) . \end{aligned}$$
(66)
It takes a nonnegative value when
$$\begin{aligned} \sum _{l=k_1+1}^{k_2} (-p_{i'_l}+p_{j'_l})\ge 0 \quad \text {or} \quad \sum _{l=k_3}^N (p_{i'_l} - p_{j'_l})\ge 0 \end{aligned}$$
(67)
holds for any \(0\le k_1<k_2<k_3\le N\) . Next let us compare (65 ) and RHS of(63 ) in the case
$$\begin{aligned} \sum _{l=k_1+1}^{k_2} (-p_{i'_l}+p_{j'_l})<0 \end{aligned}$$
(68)
and
$$\begin{aligned} \sum _{l=k_3}^N (p_{i'_l} - p_{j'_l})<0 \end{aligned}$$
(69)
hold for certain \(k_1\) , \(k_2\) and \(k_3\) . We introduce the notations
$$\begin{aligned} i'_l = {\left\{ \begin{array}{ll} 1 &{} \quad (l=1, 2, \dots , \alpha )\\ 2 &{} \quad (l=\alpha +1, \alpha +2, \dots , \beta )\\ 3 &{} \quad (l=\beta +1, \beta +2, \dots , N) \end{array}\right. }, \quad j'_l= {\left\{ \begin{array}{ll} 1 &{} \quad (l=1, 2, \dots , \gamma )\\ 2 &{} \quad (l=\gamma +1, \gamma +2, \dots , \delta )\\ 3 &{} \quad (l=\delta +1, \delta +2, \dots , N) \end{array}\right. }. \end{aligned}$$
(70)
If \(\alpha \ge k_2\) , then \(p_{i'_l}\) equals to \(p_1\) when \(l=1, 2, \dots , k_2\) and (68 ) does not hold. Thus \(\alpha \) should be \(\alpha < k_2\) . Similarly, \(\beta \) should be \(\beta \ge k_3\) for (69 ). Due to the obtained condition \(\alpha<k_2<k_3\le \beta \) , \(p_{i'_l}\) takes either \(p_1\) or \(p_2\) when \(l=1, 2, \dots , k_3\) . Hence, the condition (68 ) requires there exists \(l\in \{k_1+1, \dots , k_2 \}\) such that \(p_{i'_l}=2\) and \(p_{j'_l}=1\) . Thus \(\gamma \) should be \(\alpha <\gamma \) and \(k_1+1<\gamma \) . Similarly, \(\delta \) should be \(\delta <\beta \) for (69 ). Therefore we obtain \(\alpha< \gamma \le \delta <\beta \) , \(\alpha< k_2<k_3\le \beta \le N\) and \(k_1+1<\gamma \) . These conditions derive \(i'_{k_2}=i'_{k_3}=2\) , \(j'_{k_1+1}=j'_{\alpha }=1\) and \(j'_{\beta }=j'_N =3\) (see Table 1 ). For these \(i'_l\) and \(j'_l\) , we set \(i_l\) and \(j_l\) as
$$\begin{aligned} i_l = {\left\{ \begin{array}{ll} 1 &{} \quad (l=1, 2, \dots , \alpha +1 )\\ 2 &{} \quad (l=\alpha +2, \alpha +3, \dots , \beta )\\ 3 &{} \quad (l=\beta +1, \beta +2, \dots , N) \end{array}\right. }, \quad j_l= {\left\{ \begin{array}{ll} 1 &{} \quad (l=1, 2, \dots , \gamma -1 )\\ 2 &{} \quad (l=\gamma , \gamma +1, \dots , \delta )\\ 3 &{} \quad (l=\delta +1, \delta +2, \dots , N) \end{array}\right. }. \end{aligned}$$
(71)
Table 1 The sets of \(i_l\) , \(j_l\) , \(i'_l\) , \(j'_l\) for (70 ) and (71 )
Subtracting (65 ) associated with (70 ) from the argument of RHS of (63 ) associated with (71 ), we obtain
$$\begin{aligned}&\mathrm{up}[C_{i_1i_2\dots i_N}]+\mathrm{up}[C_{j_1j_2\dots j_N}]+\sum _{l=1}^N l p_{i_l}+ \sum _{l=1}^N lp_{j_l}-\sum _{l=1}^{k_2} p_{i_l}-\sum _{l=1}^{k_1} p_{j_l} +\sum _{l=k_3}^N p_{j_l}\nonumber \\&\qquad -\, \left( \mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] +\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] + \sum _{l=1}^N l p_{i'_l}+ \sum _{l=1}^N lp_{j'_l}-\sum _{l=1}^{k_1} p_{i'_l}-\sum _{l=1}^{k_2} p_{j'_l} +\sum _{l=k_3}^N p_{j'_l} \right) \nonumber \\&\quad =\mathrm{up}[C_{i_1i_2\dots i_N}]+\mathrm{up}[C_{j_1j_2\dots j_N}]-\mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] -\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] \nonumber \\&\qquad +\, (\gamma -\alpha -1) (p_2-p_1) -\sum _{l=1}^{k_2} (p_{i_l}-p_{j'_l})+\sum _{l=1}^{k_1} (p_{i'_l}-p_{j_l}) +\sum _{l=k_3}^N(p_{j_l}-p_{j'_l})\nonumber \\&\quad \ge \mathrm{up}[C_{i_1i_2\dots i_N}]+\mathrm{up}[C_{j_1j_2\dots j_N}]-\mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] -\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] \nonumber \\&\qquad +\, (\gamma -\alpha -1)(p_2-p_1)-(\gamma -\alpha -1) (p_2-p_1)\nonumber \\&\qquad +\,\sum _{l=1}^{k_1} (p_{i'_l}-p_{j_l}) +\sum _{l=k_3}^N(p_{j_l}-p_{j'_l}) \nonumber \\&\quad = \mathrm{up}[C_{i_1i_2\dots i_N}]+\mathrm{up}[C_{j_1j_2\dots j_N}]-\mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] \nonumber \\&\qquad -\,\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] +\sum _{l=1}^{k_1} (p_{i'_l}-p_{j_l})+\sum _{l=k_3}^N(p_{j_l}-p_{j'_l}) \end{aligned}$$
(72)
It takes a nonnegative value from Proposition 2 and \(k_1+1<\gamma \) . Therefore (35 ) holds.