\documentclass{article}
\usepackage{ifthen}
\usepackage{epsfig}
\usepackage[utf8]{inputenc}
\usepackage{xcolor}
% Packages requested by user
\usepackage{ppl}

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  \doxynewunicodechar{²}{${}^{2}$}% Superscript two
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\pagestyle{empty}
\begin{document}
$n$
\pagebreak

$ \Rset^n $
\pagebreak

$0 \geq 2$
\pagebreak

$0 \leq 2$
\pagebreak

$x \leq 2$
\pagebreak

$x - y \leq 2$
\pagebreak

$x + y \leq 2$
\pagebreak

$x = 2$
\pagebreak

$x - 3y \leq 2$
\pagebreak

$x - 3y < 2$
\pagebreak

$0 \equiv_2 1$
\pagebreak

$0 \equiv_2 2$
\pagebreak

$x + 3y \equiv_0 0$
\pagebreak

$x + 3y \equiv_5 0$
\pagebreak

$2x \leq 5$
\pagebreak

$x \leq 3$
\pagebreak

$w\textrm{-bit}$
\pagebreak

$2^w$
\pagebreak

$x \bmod 2^w$
\pagebreak

\[
      \mathrm{wrap}^\mathrm{u}_w(x)
        \defeq
          x - 2^w \lfloor x/2^w \rfloor.
    \]
\pagebreak

\[
      \mathrm{wrap}^\mathrm{s}_w(x)
        \defeq
          \begin{cases}
            \mathrm{wrap}^\mathrm{u}_w(x),
              &\text{if $\mathrm{wrap}^\mathrm{u}_w(x) < 2^{w-1}$;} \\
            \mathrm{wrap}^\mathrm{u}_w(x) - 2^w,
              &\text{otherwise.}
          \end{cases}
    \]
\pagebreak

$x$
\pagebreak

$S \sseq \Rset$
\pagebreak

$\bigl\{\, \mathrm{wrap}^\mathrm{u}_w(z) \bigm| z \in S \,\bigr\}$
\pagebreak

$\Rset^n$
\pagebreak

$n\textrm{-dimensional}$
\pagebreak

$\Rset$
\pagebreak

$\nonnegRset$
\pagebreak

$i \in \{0, \ldots, n-1\}$
\pagebreak

$v_i$
\pagebreak

$i\textrm{-th}$
\pagebreak

$\vect{v} = (v_0, \ldots, v_{n-1})^\transpose \in \Rset^n$
\pagebreak

$\vect{0}$
\pagebreak

$\vect{v} \in \Rset^n$
\pagebreak

$\Rset^{n \times 1}$
\pagebreak

$\vect{v}^\transpose$
\pagebreak

$\vect{v},\vect{w} \in \Rset^n$
\pagebreak

$\langle \vect{v}, \vect{w} \rangle$
\pagebreak

\[
  \vect{v}^\transpose \vect{w} = \sum_{i=0}^{n-1} v_i w_i.
\]
\pagebreak

$S_1, S_2 \sseq \Rset^n$
\pagebreak

$S_1$
\pagebreak

$S_2$
\pagebreak

$S_1 + S_2 = \{\, \vect{v}_1 + \vect{v}_2 \mid
                \vect{v}_1 \in S_1, \vect{v}_2 \in S_2 \,\}.$
\pagebreak

$\vect{a} \in \Rset^n$
\pagebreak

$b \in \Rset$
\pagebreak

$\vect{a} \neq \vect{0}$
\pagebreak

$\mathord{\relsym} \in \{ =, \geq, > \}$
\pagebreak

$\langle \vect{a}, \vect{x} \rangle \relsym b$
\pagebreak

$\mathord{\relsym} \in \{ = \}$
\pagebreak

$\mathord{\relsym} \in \{ \geq \}$
\pagebreak

$\mathord{\relsym} \in \{ > \}$
\pagebreak

$\langle \vect{a}, \vect{x} \rangle = b$
\pagebreak

$\langle \vect{a}, \vect{x} \rangle \geq b$
\pagebreak

$\langle -\vect{a}, \vect{x} \rangle \geq -b$
\pagebreak

$\vect{a} = \vect{0}$
\pagebreak

$\langle \vect{0}, \vect{x} \rangle \relsym b$
\pagebreak

$\emptyset$
\pagebreak

$\cP \sseq \Rset^n$
\pagebreak

$\cP$
\pagebreak

$n = 0$
\pagebreak

$\cP = \emptyset$
\pagebreak

$\Pset_n$
\pagebreak

$\cP \in \Pset_n$
\pagebreak

$\CPset_n$
\pagebreak

$\lambda \in \nonnegRset$
\pagebreak

\[
  \cP
    \sseq
      \bigl\{\,
        \vect{x} \in \Rset^n
      \bigm|
        - \lambda \leq x_j \leq \lambda \text{ for } j = 0, \ldots, n-1
      \,\bigr\}.
\]
\pagebreak

\[
  \cP
    \defeq
      \{\,
        \vect{x} \in \Rset^n
      \mid
        A_1 \vect{x} =    \vect{b}_1,
        A_2 \vect{x} \geq \vect{b}_2,
        A_3 \vect{x} >    \vect{b}_3
      \,\},
\]
\pagebreak

$i \in \{1, 2, 3\}$
\pagebreak

$A_i \in \Rset^{m_i} \times \Rset^n$
\pagebreak

$\vect{b}_i \in \Rset^{m_i}$
\pagebreak

$m_1, m_2, m_3 \in \Nset$
\pagebreak

$S = \{ \vect{x}_1, \ldots, \vect{x}_k \} \sseq \Rset^n$
\pagebreak

$\lambda_1, \ldots, \lambda_k \in \Rset$
\pagebreak

$\vect{v} = \sum_{j=1}^k \lambda_j \vect{x}_j$
\pagebreak

$S$
\pagebreak

$\forall j \in \{ 1, \ldots, k \} \itc \lambda_j \in \nonnegRset$
\pagebreak

$\sum_{j = 1}^k \lambda_j = 1$
\pagebreak

$\linearhull(S)$
\pagebreak

$\conichull(S)$
\pagebreak

$\affinehull(S)$
\pagebreak

$\convexhull(S)$
\pagebreak

$P, C \sseq \Rset^n$
\pagebreak

$P \union C = S$
\pagebreak

$\NNChull(P, C)$
\pagebreak

$\lambda_j > 0$
\pagebreak

$\vect{x}_j \in P$
\pagebreak

$P$
\pagebreak

$\NNChull(P, C) = \NNChull(P, P \union C)$
\pagebreak

$C \sseq P$
\pagebreak

\[
   \convexhull(P)
     = \NNChull(P, \emptyset)
     = \NNChull(P, P)
     = \NNChull(P, C).
\]
\pagebreak

$\vect{p} \in \cP$
\pagebreak

$\vect{c} \in \Rset^n$
\pagebreak

$\vect{r} \in \Rset^n$
\pagebreak

$\vect{r} \neq \vect{0}$
\pagebreak

$\cP \neq \emptyset$
\pagebreak

$\vect{p} + \lambda \vect{r} \in \cP$
\pagebreak

$\vect{l} \in \Rset^n$
\pagebreak

$\vect{l}$
\pagebreak

$-\vect{l}$
\pagebreak

$\vect{r}$
\pagebreak

$\vect{r}_1$
\pagebreak

$\vect{r}_2$
\pagebreak

$\vect{r} \neq \lambda \vect{r}_1$
\pagebreak

$\vect{r} \neq \lambda \vect{r}_2$
\pagebreak

$\vect{r}_1 \neq \lambda \vect{r}_2$
\pagebreak

$L$
\pagebreak

$R$
\pagebreak

$C$
\pagebreak

$\cG = (L, R, P, C)$
\pagebreak

\[
  \cP = \linearhull(L) + \conichull(R) + \NNChull(P, C),
\]
\pagebreak

$+$
\pagebreak

$\cP \in \CPset_n$
\pagebreak

$\cG = (L, R, P)$
\pagebreak

\[
  \cP = \linearhull(L) + \conichull(R) + \convexhull(P).
\]
\pagebreak

$P = \emptyset$
\pagebreak

$\Rset^2$
\pagebreak

$y \geq 0$
\pagebreak

$L = \bigl\{ (1, 0)^\transpose \bigr\}$
\pagebreak

$R = \bigl\{ (0, 1)^\transpose \bigr\}$
\pagebreak

$P = \bigl\{ (0, 0)^\transpose \bigr\}$
\pagebreak

$C = \emptyset$
\pagebreak

$\cC$
\pagebreak

$\cG' = (L', R', P', C') \neq \cG$
\pagebreak

$L' \sseq L$
\pagebreak

$R' \sseq R$
\pagebreak

$P' \sseq P$
\pagebreak

$C' \sseq C$
\pagebreak

$\cG$
\pagebreak

$(\cC, \cG)$
\pagebreak

$(L, R, P)$
\pagebreak

$n \in \Nset$
\pagebreak

$\vect{a}, \vect{x} \in \Rset^m$
\pagebreak

$m \leq n$
\pagebreak

$\vect{x} \in \Rset^m$
\pagebreak

$\{ \vect{x}_1, \ldots, \vect{x}_k \} \sseq \Rset^n$
\pagebreak

\[
  \sum_{i = 1}^k \lambda_i \vect{x}_i = \vect{0},
    \quad
  \sum_{i = 1}^k \lambda_i = 0
\]
\pagebreak

$i = 1, \ldots, k$
\pagebreak

$\lambda_i = 0$
\pagebreak

$n + 1$
\pagebreak

$k \in \Nset$
\pagebreak

$\pdim(\cP) = k$
\pagebreak

$k + 1$
\pagebreak

$0 \leq \pdim(\cP) \leq n$
\pagebreak

$-1$
\pagebreak

$k \leq n$
\pagebreak

$\pdim(\cP) \neq \pdim(\cQ)$
\pagebreak

$\cQ$
\pagebreak

$\pdim(\cP) = \pdim(\cQ)$
\pagebreak

$\cP_1, \cP_2 \in \Pset_n$
\pagebreak

$\cP_1$
\pagebreak

$\cP_2$
\pagebreak

$\cP_1 \inters \cP_2$
\pagebreak

$\cP_1 \uplus \cP_2$
\pagebreak

$\cP_1, \cP_2 \in \CPset_n$
\pagebreak

$\cQ \in \Pset_m$
\pagebreak

$\cR \in \Pset_{n+m}$
\pagebreak

\[
  \cR
    \defeq
      \Bigl\{\,
        (x_0, \ldots, x_{n-1}, y_0, \ldots, y_{m-1})^\transpose
          \in \Rset^{n+m}
      \Bigm|
        (x_0, \ldots, x_{n-1})^\transpose \in \cP,
        (y_0, \ldots, y_{m-1})^\transpose \in \cQ
      \,\Bigl\}.
\]
\pagebreak

$n+m$
\pagebreak

$i$
\pagebreak

$\cQ \in \Pset_{n+i}$
\pagebreak

$i+n$
\pagebreak

$\cP \sseq \Rset^2$
\pagebreak

\[
 \cQ = \bigl\{\,
         (x_0, x_1, x_2)^\transpose \in \Rset^3
       \bigm|
         (x_0, x_1)^\transpose \in \cP
       \,\bigr\}.
\]
\pagebreak

\[
  \cQ = \bigl\{\,
          (x_0, x_1, 0)^\transpose \in \Rset^3
        \bigm|
          (x_0, x_1)^\transpose \in \cP
        \,\bigr\}.
\]
\pagebreak

$\cP \in \Pset_4$
\pagebreak

$\bigl\{ (3, 1, 0, 2)^\transpose \bigr\} \sseq \Rset^4$
\pagebreak

$\{x_1, x_2\}$
\pagebreak

\[
  \cQ = \bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2.
\]
\pagebreak

$m$
\pagebreak

$m = 2$
\pagebreak

\[
  \cQ = \bigl\{ (3, 1)^\transpose \bigr\} \sseq \Rset^2.
\]
\pagebreak

$\pard{\rho}{\{0, \ldots, n-1\}}{\Nset}$
\pagebreak

$\rho\bigl(\{0, \ldots, n-1\}\bigr) = \{0, \ldots, m-1\}$
\pagebreak

$\rho$
\pagebreak

$m = 0$
\pagebreak

$\Rset^0$
\pagebreak

\[
  \cQ
    \defeq
      \Bigl\{\,
        \bigl(v_{\rho^{-1}(0)}, \ldots, v_{\rho^{-1}(m-1)}\bigr)^\transpose
      \Bigm|
        (v_0, \ldots, v_{n-1})^\transpose \in \cP
      \,\Bigr\}.
\]
\pagebreak

$n > 0$
\pagebreak

$n+1$
\pagebreak

$\ldots$
\pagebreak

$n+m-1$
\pagebreak

\[
  \cQ
    \defeq
      \sset{
        \vect{u} \in \Rset^{n+m}
      }{
        \exists \vect{v}, \vect{w} \in \cP
          \st u_i = v_i \\
          \qquad \mathord{} \land \forall j = n, n+1, \ldots, n+m-1
                                     \itc u_j = w_i \\
          \qquad \mathord{} \land \forall k = 0, \ldots, n-1
                                     \itc k \neq i \implies u_k = v_k = w_k
      }.
\]
\pagebreak

$J = \{ j_0, \ldots, j_{m-1} \}$
\pagebreak

$m < n$
\pagebreak

$j < n$
\pagebreak

$j \in J$
\pagebreak

$i < n$
\pagebreak

$i \notin J$
\pagebreak

\[
  \cQ \defeq \biguplus_{d = 0}^m \cQ_d
\]
\pagebreak

\[
  \cQ_m
    \defeq
      \sset{
        \vect{u} \in \Rset^{n-m}
      }{
        \exists \vect{v} \in \cP
          \st u_{i'} = v_i \\
          \qquad \mathord{} \land \forall k = 0, \ldots, n-1
                                     \itc k \neq i \implies u_{k'} = v_k
      }
\]
\pagebreak

$ d = 0 $
\pagebreak

$ \ldots $
\pagebreak

$ m-1 $
\pagebreak

\[
  \cQ_d
    \defeq
      \sset{
        \vect{u} \in \Rset^{n-m}
      }{
        \exists \vect{v} \in \cP
          \st u_{i'} = v_{j_d} \\
          \qquad \mathord{} \land \forall k = 0, \ldots, n-1
                                     \itc k \neq i \implies u_{k'} = v_k
      },
\]
\pagebreak

$ k = 0 $
\pagebreak

$ n-1 $
\pagebreak

\[
  k'
    \defeq
      k - \card \{\, j \in J \mid k > j \,\},
\]
\pagebreak

$\card S$
\pagebreak

$\reld{\phi}{\Rset^n}{\Rset^m}$
\pagebreak

$\phi(S) \sseq \Rset^m$
\pagebreak

$\phi$
\pagebreak

$S \sseq \Rset^n$
\pagebreak

\[
  \phi(S)
    \defeq
      \bigl\{\,
        \vect{w} \in \Rset^m
      \bigm|
        \exists \vect{v} \in S \st
          (\vect{v}, \vect{w}) \in \phi
      \,\bigr\}.
\]
\pagebreak

$\phi^{-1}(S') \sseq \Rset^n$
\pagebreak

$S' \sseq \Rset^m$
\pagebreak

\[
  \phi^{-1}(S')
    \defeq
      \bigl\{\,
        \vect{v} \in \Rset^n
      \bigm|
        \exists \vect{w} \in S' \st
          (\vect{v}, \vect{w}) \in \phi
      \,\bigr\}.
\]
\pagebreak

$n = m$
\pagebreak

$\ell \in \Nset$
\pagebreak

\[
  \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^m
    \itc
      (\vect{v}, \vect{w}) \in \phi
        \iff
          \bigland_{i=1}^{\ell}
            \bigl(
              \langle \vect{c}_i, \vect{w} \rangle
                \relsym_i
                  \langle \vect{a}_i, \vect{v} \rangle + b_i
            \bigr),
\]
\pagebreak

$\vect{a}_i \in \Rset^n$
\pagebreak

$\vect{c}_i \in \Rset^m$
\pagebreak

$b_i \in \Rset$
\pagebreak

$\mathord{\relsym}_i \in \{ <, \leq, =, \geq, > \}$
\pagebreak

$i = 1, \ldots, \ell$
\pagebreak

$A \in \Rset^m \times \Rset^n$
\pagebreak

$\vect{b} \in \Rset^m$
\pagebreak

\[
  \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^m
    \itc
      (\vect{v}, \vect{w}) \in \phi
        \iff
          \vect{w} = A\vect{v} + \vect{b}.
\]
\pagebreak

$<$
\pagebreak

$>$
\pagebreak

$\reld{\phi}{\Rset^n}{\Rset^n}$
\pagebreak

$\vect{x} = (x_0, \ldots, x_{n-1})^\transpose$
\pagebreak

$\vect{x}' = (x'_0, \ldots, x'_{n-1})^\transpose$
\pagebreak

$x'_i$
\pagebreak

$x'_i = x_i$
\pagebreak

$\reld{\phi}{\Rset^3}{\Rset^3}$
\pagebreak

$x'_0 - x'_2 \geq 2 x_0 - x_1$
\pagebreak

$x'_1$
\pagebreak

\[
  \forall \vect{v} \in \Rset^3, \vect{w} \in \Rset^3
    \itc
      (\vect{v}, \vect{w}) \in \phi
        \iff
          (w_0 - w_2 \geq 2 v_0 - v_1) \land (w_1 = v_1).
\]
\pagebreak

$x'_0 + 0 \cdot x'_1 - x'_2 \geq 2 x_0 - x_1$
\pagebreak

$x'_k$
\pagebreak

$\langle \vect{a}, \vect{x} \rangle + b$
\pagebreak

$\fund{\phi = \bigl(x'_k = \langle \vect{a}, \vect{x} \rangle + b\bigr)}
        {\Rset^n}{\Rset^n}$
\pagebreak

\[
  \forall \vect{v} \in \Rset^n \itc
    \phi(\vect{v}) = A\vect{v} + \vect{b},
\]
\pagebreak

\[
 A =
   \begin{pmatrix}
      1      &        &    0    &   0    & \cdots  & \cdots &    0    \\
             & \ddots &         & \vdots &         &        & \vdots  \\
      0      &        &    1    &   0    & \cdots  & \cdots &    0    \\
     a_0     & \cdots & a_{k-1} &  a_k   & a_{k+1} & \cdots & a_{n-1} \\
      0      & \cdots & \cdots  &   0    &   1     &        &    0    \\
    \vdots   &        &         & \vdots &         & \ddots &         \\
      0      & \cdots & \cdots  &   0    &   0     &        &    1
   \end{pmatrix}, \qquad
 \vect{b} =
   \begin{pmatrix}
     0      \\
     \vdots \\
     0      \\
     b      \\
     0      \\
     \vdots \\
     0
   \end{pmatrix}
\]
\pagebreak

$a_i$
\pagebreak

$b$
\pagebreak

$(k+1)$
\pagebreak

$A$
\pagebreak

$\vect{b}$
\pagebreak

$(v_0, \ldots, v_{n-1})^\transpose$
\pagebreak

\[
   \Bigl(v_0, \ldots,
           \bigl(\textstyle{\sum_{i=0}^{n-1}} a_i v_i + b\bigr),
             \ldots, v_{n-1}\Bigr)^\transpose.
\]
\pagebreak

$x'_k = \langle \vect{a}, \vect{x} \rangle + b$
\pagebreak

$\bigl\{
     (0, 0)^\transpose,
     (0, 3)^\transpose,
     (3, 0)^\transpose,
     (3, 3)^\transpose
   \bigr\}$
\pagebreak

$x_0$
\pagebreak

$x_0 + 2 x_1 + 4$
\pagebreak

$k = 0$
\pagebreak

$a_0 = 1, a_1 = 2, b = 4$
\pagebreak

$\bigl\{
     (4, 0)^\transpose,
     (10, 3)^\transpose,
     (7, 0)^\transpose,
     (13, 3)^\transpose
   \bigr\}$
\pagebreak

$x_0 - 2 x_1$
\pagebreak

$x_1$
\pagebreak

$a_0 = 0, a_1 = 1, b = 0$
\pagebreak

$(1, 0)^\transpose$
\pagebreak

$a_k$
\pagebreak

$\mathrm{lb} = \langle \vect{a}, \vect{x} \rangle + b$
\pagebreak

$\mathrm{ub} = \langle \vect{c}, \vect{x} \rangle + d$
\pagebreak

$\phi = (\mathrm{lb} \leq x'_k \leq \mathrm{ub})$
\pagebreak

\[
  \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n
    \itc
      (\vect{v}, \vect{w}) \in \phi
        \iff
          \bigl(
            \langle \vect{a}, \vect{v} \rangle + b
              \leq w_k
                \leq \langle \vect{c}, \vect{v} \rangle + d
          \bigr)
            \land
          \Bigl(
            \bigland_{0 \leq i < n, i \neq k}
              w_i = v_i
          \Bigr).
\]
\pagebreak

$\mathbb{F}_{\mathrm{f}}$
\pagebreak

$\mathrm{f}$
\pagebreak

$\mathbb{I}_\mathrm{f}$
\pagebreak

$\langle\vect{\alpha}, \vect{x}\rangle
 + \beta$
\pagebreak

\[\left<\vect{\alpha}, \vect{x}\right> + \beta =
        \alpha_0x_0 + \ldots +
        \alpha_{n-1}x_{n-1} + \beta,\]
\pagebreak

$\beta$
\pagebreak

$\vect{\alpha}_k \in \mathbb{I}_{\mathrm{f}}$
\pagebreak

$k = 0, \ldots, n - 1$
\pagebreak

$\mathrm{lf}$
\pagebreak

$\mathrm{lb} \leq x'_k \leq \mathrm{ub}$
\pagebreak

$\mathrm{lb}$
\pagebreak

$\mathrm{ub}$
\pagebreak

$\phi = (\mathrm{lhs}' \relsym \mathrm{rhs})$
\pagebreak

$\mathrm{lhs} = \langle \vect{c}, \vect{x} \rangle + d$
\pagebreak

$\mathrm{rhs} = \langle \vect{a}, \vect{x} \rangle + b$
\pagebreak

$\mathord{\relsym} \in \{ <, \leq, =, \geq, > \}$
\pagebreak

\[
  \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n
    \itc
      (\vect{v}, \vect{w}) \in \phi
        \iff
          \bigl(
            \langle \vect{c}, \vect{w} \rangle + d
              \relsym \langle \vect{a}, \vect{v} \rangle + b
          \bigr)
            \land
          \Bigl(
            \bigland_{0 \leq i < n, c_i = 0}
              w_i = v_i
          \Bigr).
\]
\pagebreak

$\mathrm{lhs} = x_k$
\pagebreak

$x'_k = \mathrm{rhs}$
\pagebreak

$\mathrm{lhs}$
\pagebreak

$\mathrm{rhs}$
\pagebreak

$\mathrm{lhs}' \leq \mathrm{rhs}$
\pagebreak

$\mathrm{rhs}' \geq \mathrm{lhs}$
\pagebreak

$\cQ \in \Pset_n$
\pagebreak

$i \in \{ 0, \ldots, n-1 \}$
\pagebreak

\[
  \cQ
    = \bigl\{\,
        \vect{w} \in \Rset^n
      \bigm|
        \exists \vect{v} \in \cP \st
          \forall j \in \{0, \ldots, n-1\} \itc
            j \neq i \implies w_j = v_j
      \,\bigr\}.
\]
\pagebreak

$\cP, \cQ \in \Pset_n$
\pagebreak

$ \cP \nearrow \cQ$
\pagebreak

\[
  \bigl\{\,
    \vect{p} + \lambda \vect{q} \in \Rset^n
  \bigm|
    \vect{p} \in \cP, \vect{q} \in \cQ, \lambda \in \nonnegRset
  \,\bigr\}.
\]
\pagebreak

$ \cP
\nearrow_{>0} \cQ$
\pagebreak

\[
  \bigl\{\,
    \vect{p} + \lambda \vect{q} \in \Rset^n
  \bigm|
    \vect{p} \in \cP, \vect{q} \in \cQ, \lambda \in \Rset^{>0}
  \,\bigr\},
\]
\pagebreak

$\Rset^{>0}$
\pagebreak

$ \cP \nearrow_{>0} \cQ$
\pagebreak

$\cP \nearrow \cQ$
\pagebreak

$\cP \nearrow \cQ = \cP \cup \left(\cP \nearrow_{>0} \cQ\right)$
\pagebreak

$\cP, \cQ, \cR \in \Pset_n$
\pagebreak

$\cR$
\pagebreak

$\cR \inters \cQ = \cP \inters \cQ$
\pagebreak

$\cR \Sseq \cP$
\pagebreak

$r \leq p$
\pagebreak

$r$
\pagebreak

$p$
\pagebreak

$
  c = \bigl(
        \langle \vect{a}, \vect{x} \rangle \relsym b
      \bigr)
$
\pagebreak

$c$
\pagebreak

$\cP \inters \cQ = \emptyset$
\pagebreak

$\cP \inters \cQ \neq \emptyset$
\pagebreak

$\cP \inters \cQ \subset \cP$
\pagebreak

$\cP \sseq \cQ$
\pagebreak

$\cP \sseq \cH$
\pagebreak

$\cH$
\pagebreak

$g$
\pagebreak

$\cP \widen \cQ$
\pagebreak

$\cP, \cQ \in \CPset_n$
\pagebreak

$\cP \Sseq \cP \widen \cQ$
\pagebreak

$\cQ \sseq \cP$
\pagebreak

$\cP = \cQ$
\pagebreak

$k$
\pagebreak

$\mathord{\relsym} \in \{ =, \geq, >\}$
\pagebreak

$k \in \{ 0, \ldots, i-1, i+1, \ldots, n-1 \}$
\pagebreak

$a_k = 0$
\pagebreak

$x = r$
\pagebreak

$x \leq r$
\pagebreak

$x \geq r$
\pagebreak

$x < r$
\pagebreak

$x > r$
\pagebreak

$r \in \Rset$
\pagebreak

$\cB$
\pagebreak

$\vect{e}_i = (0, \ldots, 1, \ldots, 0)^\transpose$
\pagebreak

$\langle \vect{e}_i, \vect{x} \rangle \relsym b$
\pagebreak

$\mathord{\relsym}$
\pagebreak

$\mathord{\geq}$
\pagebreak

$\mathord{>}$
\pagebreak

$\langle\vect{e}_i,\vect{x}\rangle \relsym b$
\pagebreak

$\mathord{\leq}$
\pagebreak

$\mathord{<}$
\pagebreak

$\{-2, -1, 0, 1, 2\}$
\pagebreak

$\mathord{\relsym} \in \{ =, \geq\}$
\pagebreak

$i, j \in \{ 0, \ldots, n-1 \}$
\pagebreak

$a_i, a_j \in \{ -1, 0, 1 \}$
\pagebreak

$a_i \neq a_j$
\pagebreak

$k \notin \{ i, j \}$
\pagebreak

$a, b, f \in \Rset$
\pagebreak

$a \equiv_f b$
\pagebreak

$\exists \mu \in \Zset \st a - b = \mu f$
\pagebreak

$\Sset \in \{ \Qset, \Rset \}$
\pagebreak

$\vect{a} \in \Sset^n \setdiff \{\vect{0}\}$
\pagebreak

$b, f \in \Sset$
\pagebreak

$\langle \vect{a}, \vect{x} \rangle \equiv_f b$
\pagebreak

$\Sset^n$
\pagebreak

\[
  \bigl\{\,
    \vect{v} \in \Rset^n
  \bigm|
    \exists \mu \in \Zset \st
      \langle \vect{a}, \vect{v} \rangle = b + \mu f
  \,\bigr\};
\]
\pagebreak

$f \neq 0$
\pagebreak

$\langle \vect{a}, \vect{x} \rangle \equiv_0 b$
\pagebreak

$f = 0$
\pagebreak

$f$
\pagebreak

\[
  \big\{\,
    \bigl(\langle \vect{a}, \vect{x} \rangle = b + \mu f\bigr)
  \bigm|
    \mu \in \Zset
  \,\bigr\};
\]
\pagebreak

$b \equiv_f 0$
\pagebreak

$\langle \vect{0}, \vect{x} \rangle \equiv_f b$
\pagebreak

$\cL  \sseq \Rset^n$
\pagebreak

$\cL$
\pagebreak

$\Qset^n$
\pagebreak

$\cL = \emptyset$
\pagebreak

$\Gset_{n}$
\pagebreak

$\bigl\{\langle\vect{0}, \vect{x}\rangle \equiv_0 1\bigr\}$
\pagebreak

$0 = 1$
\pagebreak

$\bigl\{\langle\vect{a}, \vect{x}\rangle \equiv_2 0,
     \langle\vect{a}, \vect{x}\rangle \equiv_2 1\bigr\}$
\pagebreak

$\Gset_n$
\pagebreak

$\mu_1, \ldots, \mu_k \in \Zset$
\pagebreak

$\vect{v} = \sum_{j=1}^k \mu_j \vect{x}_j$
\pagebreak

$\inthull(S)$
\pagebreak

$\intaffinehull(S)$
\pagebreak

$\vect{p} \in \cL$
\pagebreak

$\vect{q} \in \Rset^n$
\pagebreak

$\vect{q} \neq \vect{0}$
\pagebreak

$\cL \neq \emptyset$
\pagebreak

$\vect{p} + \mu \vect{q} \in \cL$
\pagebreak

$\mu \in \Zset$
\pagebreak

$\vect{p} + \lambda \vect{l} \in \cL$
\pagebreak

$\lambda \in \Rset$
\pagebreak

$L, Q, P$
\pagebreak

\[
  \cL = \linearhull(L) + \inthull(Q) + \intaffinehull(P)
\]
\pagebreak

$\cL \in \Gset_n$
\pagebreak

$(L, Q, P)$
\pagebreak

$\cL = \ggen(L, Q, P)$
\pagebreak

$\cL = \ggen(L, Q, P) = \emptyset$
\pagebreak

$P \neq \emptyset$
\pagebreak

$\cL = \ggen(L, \emptyset, Q_{\vect{p}} \union P)$
\pagebreak

$\vect{p} \in P$
\pagebreak

$Q_{\vect{p}} = \{\, \vect{p} + \vect{q} \mid \vect{q} \in Q \,\}$
\pagebreak

$\cC'$
\pagebreak

$\card \cC \leq \card \cC'$
\pagebreak

$\cG = (L, Q, P)$
\pagebreak

$\cG' = (L', Q', P')$
\pagebreak

$\card L \leq \card L'$
\pagebreak

$\card Q + \card P \leq \card Q' + \card P'$
\pagebreak

$\pdim(\cG) = k$
\pagebreak

$0 \leq \pdim(\cG) \leq n$
\pagebreak

$x_k$
\pagebreak

$\mathrm{expr} = \langle \vect{a}, \vect{x} \rangle + b$
\pagebreak

\[
   \Bigl(v_0, \ldots, \bigl(\textstyle{\sum_{i=0}^{n-1}} a_i v_i + b\bigr),
                   \ldots, v_{n-1}\Bigr)^\transpose.
\]
\pagebreak

$\bigl\{
     (0, 0)^\transpose,
     (0, 3)^\transpose,
     (3, 0)^\transpose
   \bigr\}$
\pagebreak

$3x_0 + 2 x_1 + 1$
\pagebreak

$a_0 = 3, a_1 = 2, b = 1$
\pagebreak

$\cL_1$
\pagebreak

$\bigl\{
     (1, 0)^\transpose,
     (7, 3)^\transpose,
     (10, 0)^\transpose
   \bigr\}$
\pagebreak

$(1, 0)$
\pagebreak

$(3, -3), (0, 9)$
\pagebreak

$\{x \equiv_3 1, x + y \equiv_9 1\}$
\pagebreak

$\cL_2$
\pagebreak

$x = y$
\pagebreak

$y$
\pagebreak

$\fund{\phi = (\mathrm{lhs}', \mathrm{rhs}, f)}{\Rset^n}{\Rset^n}$
\pagebreak

$f \in \Qset$
\pagebreak

\[
  \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n
    \itc
      (\vect{v}, \vect{w}) \in \phi
        \iff
          \bigl(
            \langle \vect{c}, \vect{w} \rangle + d
              \equiv_f \langle \vect{a}, \vect{v} \rangle + b
          \bigr)
            \land
          \Bigl(
            \bigland_{0 \leq i < n, c_i = 0}
              w_i = v_i
          \Bigr).
\]
\pagebreak

$\mathrm{expr} = \bigl(\langle \vect{a}, \vect{x} \rangle + b\bigr)$
\pagebreak

$c, f \in \Rset$
\pagebreak

$\cg = ( \mathrm{expr} \equiv_f c )$
\pagebreak

$\mathrm{expr}$
\pagebreak

$\mathrm{val} = \langle \vect{a}, \vect{w} \rangle + b$
\pagebreak

$\vect{w} \in \cL$
\pagebreak

\[
  \lvert\mathrm{val}\rvert
     = \min\Bigl\{\,
             \bigl\lvert\langle \vect{a}, \vect{v} \rangle + b \bigr\rvert
           \Bigm|
             \vect{v} \in \cL
           \,\Bigr\}.
\]
\pagebreak

$( \mathrm{expr} = c )$
\pagebreak

$0$
\pagebreak

$\cL_1, \cL_2 \in \Gset_n$
\pagebreak

$ \cL_1 \nearrow \cL_2$
\pagebreak

\[
  \bigl\{\,
    \vect{p} + \mu \vect{q} \in \Rset^n
  \bigm|
    \vect{p} \in \cL_1, \vect{q} \in \cL_2, \mu \in \Zset
  \,\bigr\}.
\]
\pagebreak

$
  \cg = \bigl(
          \langle \vect{a}, \vect{x} \rangle \equiv_f b
        \bigr)
$
\pagebreak

$\cL_{\cg} = \gcon\bigl(\{\cg\}\bigr)$
\pagebreak

$\cg$
\pagebreak

$\cL \inters \cL_{\cg} = \emptyset$
\pagebreak

$\cL \inters \cL_{\cg} \neq \emptyset$
\pagebreak

$\cL \inters \cL_{\cg} \subset \cL$
\pagebreak

$\cL \sseq \cL_{\cg}$
\pagebreak

$\cL \inters \cQ = \emptyset$
\pagebreak

$\cL \inters \cQ \neq \emptyset$
\pagebreak

$\cL \inters \cQ \subset \cL$
\pagebreak

$\cL \sseq \cQ$
\pagebreak

$=$
\pagebreak

$J$
\pagebreak

$\{0, \ldots, n-1\}$
\pagebreak

$w$
\pagebreak

$R = \{r \in \Rset \mid 0 \leq r < 2^w\}$
\pagebreak

$R = \{r \in \Rset \mid -2^{w-1} \leq r < 2^{w-1}\}$
\pagebreak

$v_j$
\pagebreak

$j$
\pagebreak

$\cL'$
\pagebreak

$a \in R$
\pagebreak

$a$
\pagebreak

$\cL' = \cL$
\pagebreak

$a + z \in R$
\pagebreak

$z \in \Zset$
\pagebreak

$(0, \ldots, 0, v_j, 0, \ldots, 0)$
\pagebreak

$v_j = 1$
\pagebreak

$v_j = a \mod 2^w$
\pagebreak

$a \in \Rset$
\pagebreak

$a'$
\pagebreak

$a' = a \mod 2^w$
\pagebreak

$a'\in R$
\pagebreak

$v_j = 2^w$
\pagebreak

$\cL_1 \widen \cL_2$
\pagebreak

$\cL_1 \sseq \cL_2$
\pagebreak

$D$
\pagebreak

$\mathord{\entails}$
\pagebreak

$\mathord{\meet}$
\pagebreak

$\true$
\pagebreak

$\false$
\pagebreak

$\cS \in \wp(D)$
\pagebreak

$\false \notin \cS$
\pagebreak

$\forall d_1, d_2 \in \cS \itc d_1 \entails d_2 \implies d_1 = d_2$
\pagebreak

$\wpfn{D}{\entails}$
\pagebreak

$\fund{\nonredmap}{\wpf(D)}{\wpfn{D}{\entails}}$
\pagebreak

$\cS \in \wpf(D)$
\pagebreak

\[
  \nonredmap(\cS)
    \defeq
      \cS \setdiff \{\,
                   d \in \cS
                 \mid
                   d = \false \text{ or } \exists d' \in \cS \st d \sentails d'
                 \,\}.
\]
\pagebreak

$d \sentails d'$
\pagebreak

$d \entails d' \land d \ne d'$
\pagebreak

$\cS$
\pagebreak

$\nonredmap(\cS)$
\pagebreak

$\cS \neq \{ \false \}$
\pagebreak

$D_{\smallP}$
\pagebreak

$\mathord{\entailsP}$
\pagebreak

$\cS_1$
\pagebreak

$\cS_2 \in D_{\smallP}$
\pagebreak

$\cS_1 \entailsP \cS_2$
\pagebreak

\[
  \forall d_1 \in \cS_1 \itc
    \exists d_2 \in \cS_2 \st
      d_1 \entails d_2.
\]
\pagebreak

$\{\true\}$
\pagebreak

$
  \nonredmap
    \bigl(
      \{\,
        d_1 \meet d_2
      \mid
        d_1 \in \cS_1, d_2 \in \cS_2
      \,\}
    \bigr)
$
\pagebreak

$\nonredmap(\cS_1 \union \cS_2)$
\pagebreak

$\cS \in D_{\smallP}$
\pagebreak

$d \in D$
\pagebreak

$\nonredmap\bigl(\cS \union \{d\}\bigr)$
\pagebreak

$\cS_1 = \{ d_1, \ldots, d_m \}$
\pagebreak

$\cS_2 = \{ c_1, \ldots, c_n \}$
\pagebreak

$\cS = \{ s_1, \ldots, s_q \}$
\pagebreak

$\cS_2$
\pagebreak

$q \leq m$
\pagebreak

$s_k \in \cS$
\pagebreak

$d_i \in \cS_1$
\pagebreak

$c_j \in \cS_2$
\pagebreak

$s_k$
\pagebreak

$d_i$
\pagebreak

$c_j$
\pagebreak

$\cS_1, \cS_2$
\pagebreak

$\uplus$
\pagebreak

$c, d$
\pagebreak

$c \uplus d \neq c \union d$
\pagebreak

$\cV$
\pagebreak

$\mathbb{F}_a$
\pagebreak

$\mathbb{F}_t$
\pagebreak

$-\infty$
\pagebreak

$+\infty$
\pagebreak

\[
  i + \sum_{v \in \cV}i_{v}v,
\]
\pagebreak

$v$
\pagebreak

$i_{v}$
\pagebreak

$\mathbb{I}_a$
\pagebreak

$\infty$
\pagebreak

$\mathbb{L}_a$
\pagebreak

\[

  \left(i + \sum_{v \in V}i_{v}v\right)
  \aslf
  \left(i' + \sum_{v \in V}i'_{v}v\right)
  \defeq
  \left(i \asifp i'\right) +
  \sum_{v \in V}\left(i_{v} \asifp i'_{v}\right)v,

\]
\pagebreak

\[

  \left(i + \sum_{v \in V}i_{v}v\right)
  \adlf
  \left(i' + \sum_{v \in V}i'_{v}v\right)
  \defeq
  \left(i \adifp i'\right) +
  \sum_{v \in V}\left(i_{v} \adifp i'_{v}\right)v,

\]
\pagebreak

\[

  i
  \amlf
  \left(i' + \sum_{v \in V}i'_{v}v\right)
  \defeq
  \left(i \amifp i'\right) +
  \sum_{v \in V}\left(i \amifp i'_{v}\right)v,

\]
\pagebreak

\[

  \left(i + \sum_{v \in V}i_{v}v\right)
  \adivlf
  i'
  \defeq
  \left(i \adivifp i'\right) +
  \sum_{v \in V}\left(i_{v} \adivifp i'\right)v.

\]
\pagebreak

$\asifp, \adifp, \amifp,$
\pagebreak

$\adivifp$
\pagebreak

$\fund{\rho^{\#}}{\cV}{\mathbb{I}_a}$
\pagebreak

$\fund{\rho^{\#}_l}{\cV}{\mathbb{L}_a}$
\pagebreak

$e$
\pagebreak

$\left \llbracket \rho^{\#}, \rho^{\#}_l \right \rrbracket$
\pagebreak

$\linexprenv{e}{\rho^{\#}}{\rho^{\#}_l}$
\pagebreak

$i+1$
\pagebreak

\[
  \sum_{i=0}^{n-1} a_i x_i + b
\]
\pagebreak

$x_i$
\pagebreak

$i + 1$
\pagebreak

$lf = x_5 - x_2 + 1$
\pagebreak

$4x - 2y - z + 14$
\pagebreak

$3$
\pagebreak

$\sum_{i=0}^{n-1} a_i x_i + b = 0$
\pagebreak

$\sum_{i=0}^{n-1} a_i x_i + b \geq 0$
\pagebreak

$\sum_{i=0}^{n-1} a_i x_i + b > 0$
\pagebreak

$3x + 5y - z = 0$
\pagebreak

$4x \geq 2y - 13$
\pagebreak

$2$
\pagebreak

$4x > 2y - 13$
\pagebreak

$x - 5y + 3z \leq 4$
\pagebreak

$x - 5y + 3z > 4$
\pagebreak

$\vect{l} = (a_0, \ldots, a_{n-1})^\transpose$
\pagebreak

$\vect{r} = (a_0, \ldots, a_{n-1})^\transpose$
\pagebreak

$\vect{p} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\pagebreak

$\vect{c} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\pagebreak

$d > 0$
\pagebreak

$x-y-z$
\pagebreak

$\vect{p} = (1, 0, 2)^\transpose \in \Rset^3$
\pagebreak

$\vect{0} \in \Rset^3$
\pagebreak

$\vect{0} \in \Rset^2$
\pagebreak

$\vect{0} \in \Rset^0$
\pagebreak

$\vect{p}$
\pagebreak

$\vect{q} = (-1.5, 3.2, 2.1)^\transpose \in \Rset^3$
\pagebreak

$\vect{c} = (1, 0, 2)^\transpose \in \Rset^3$
\pagebreak

$(a_0, \ldots, a_{n-1})^\transpose$
\pagebreak

$(a_0, 2 a_1, \ldots, (i+1)a_i, \ldots, n a_{n-1})^\transpose$
\pagebreak

$\vect{q} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\pagebreak

$\vect{p} = (1, -1, -1)^\transpose \in \Rset^3$
\pagebreak

$\vect{p1} = (-1.5, 3.2, 2.1)^\transpose \in \Rset^3$
\pagebreak

$\vect{q} = (1, 0, 2)^\transpose \in \Rset^3$
\pagebreak

$\cg = \sum_{i=0}^{n-1} a_i x_i + b = 0 \pmod m$
\pagebreak

$m \neq 0$
\pagebreak

$\pmod{1}$
\pagebreak

$4x = 2y - 13 \pmod{1}$
\pagebreak

$4x = 2y - 13 \pmod{2}$
\pagebreak

$x - 5y + 3z = 4 \pmod{5}$
\pagebreak

$2 e$
\pagebreak

$2x - 10y + 6z = 8 \pmod{5}$
\pagebreak

$1$
\pagebreak

$x + y = 1 \pmod{2}$
\pagebreak

$\Zset^2$
\pagebreak

$(0, 1)^\transpose \in \Rset^2$
\pagebreak

\[
    \bigl\{\,
      (x, 0)^\transpose \in \Rset^2
    \bigm|
      x \geq 0
    \,\bigr\},
  \]
\pagebreak

$R^0$
\pagebreak

$\{ 2 \} \sseq \Rset$
\pagebreak

\[
  \bigl\{\,
    (2, y)^\transpose \in \Rset^2
  \bigm|
    y \in \Rset
  \,\bigr\}.
\]
\pagebreak

$\bigl\{ (2, 0)^\transpose \bigr\} \sseq \Rset^2$
\pagebreak

$x+4$
\pagebreak

$x+y$
\pagebreak

$x-y$
\pagebreak

$\bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2$
\pagebreak

$\bigl\{(3, 0)^\transpose \bigr\} \sseq \Rset^2$
\pagebreak

\[
    \bigl\{\,
      (x, 0)^\transpose \in \Rset^2
    \bigm|
      x \pmod{1}\ 0
    \,\bigr\},
  \]
\pagebreak

$x + y = 2$
\pagebreak

\[
\bigl\{\,
(2, y)^\transpose \in \Rset^2
\bigm|
y \in \Rset
\,\bigr\}.
\]
\pagebreak

$x+3$
\pagebreak

$(x, y)$
\pagebreak

$x=y$
\pagebreak

$x=-y$
\pagebreak

$\pm x_i \leq k$
\pagebreak

$x_i - x_j \leq k$
\pagebreak

\[
    a_i x_i - a_j x_j \relsym b
  \]
\pagebreak

$\mathord{\relsym} \in \{ \leq, =, \geq \}$
\pagebreak

$a_j$
\pagebreak

$a_i = 0$
\pagebreak

$a_j = 0$
\pagebreak

$a_i = a_j$
\pagebreak

$3x - 3y \leq 1$
\pagebreak

$x - y \leq 1$
\pagebreak

$x - y \leq \frac{1}{3}$
\pagebreak

$x - y \leq k$
\pagebreak

$k > \frac{1}{3}$
\pagebreak

$\frac{1}{3}$
\pagebreak

$3x - y \leq 1$
\pagebreak

$\Rset^3$
\pagebreak

\[
  ax_i + bx_j \leq k
\]
\pagebreak

$a, b \in \{-1, 0, 1\}$
\pagebreak

\[
    \pm a_i x_i \pm a_j x_j \relsym b
  \]
\pagebreak

$3x + 3y \leq 1$
\pagebreak

$x + y \leq 1$
\pagebreak

$x + y \leq \frac{1}{3}$
\pagebreak

$x + y \leq k$
\pagebreak

$\perp$
\pagebreak

$M$
\pagebreak

$M-x'_i$
\pagebreak

$x'$
\pagebreak

$x'_i = M-x_i$
\pagebreak

$x_i = M-x'_i$
\pagebreak

$(x,y)$
\pagebreak

\[\left\{\begin{array}{l}
    y \geq 2x - 4\\
    y \leq -x + p
  \end{array}\right.\]
\pagebreak

\[\left\{\begin{array}{l}
    M - y \geq 2M - 2x - 4\\
    M - y \leq -M + x + p
  \end{array}\right.\]
\pagebreak

\[ \left\{\begin{array}{l}
  x' = M - \left\lfloor\frac{p+1}{3}\right\rfloor - 1 \\
  y' = M - p + \left\lfloor\frac{p+1}{3}\right\rfloor + 1
\end{array}\right.
\]
\pagebreak

\[ \left\{\begin{array}{l}
  x = \left\lfloor\frac{p+1}{3}\right\rfloor + 1 \\
  y = p - \left\lfloor\frac{p+1}{3}\right\rfloor - 1
\end{array}\right.
\]
\pagebreak

$x'_i-M$
\pagebreak

$x'_i = x_i+M$
\pagebreak

$x_i = x'_i-M$
\pagebreak

\[\left\{\begin{array}{l}
    y \geq -2x - 4\\
    2y \leq x + 2p
  \end{array}\right.\]
\pagebreak

\[\left\{\begin{array}{l}
    y' - M \geq -2x' + 2M - 4\\
    2y' - 2M \leq x' - M + 2p
  \end{array}\right.\]
\pagebreak

\[ \left\{\begin{array}{l}
  x' = M - \left\lfloor\frac{2p+3}{5}\right\rfloor - 1 \\
  y' = M + 2\left\lfloor\frac{2p+3}{5}\right\rfloor - 2
\end{array}\right.
\]
\pagebreak

\[ \left\{\begin{array}{l}
  x = -\left\lfloor\frac{2p+3}{5}\right\rfloor - 1 \\
  y = 2\left\lfloor\frac{2p+3}{5}\right\rfloor - 2
\end{array}\right.
\]
\pagebreak

$p^+-p^-$
\pagebreak

$p^+$
\pagebreak

$p^-$
\pagebreak

$p_i$
\pagebreak

$p^+_i-p^-$
\pagebreak

$-p^-$
\pagebreak

$x_1, x_2, \dots, x_n$
\pagebreak

$p_1, p_2, \dots, p_m$
\pagebreak

$f(x_2, \dots, x_n, p_1, \dots, p_m)$
\pagebreak

$x_1 \geq f(x_2, \dots, x_n, p_1, \dots, p_m)$
\pagebreak

$D_1$
\pagebreak

$D_2$
\pagebreak

$\fund{\gamma_1}{D_1}{\Rset^n}$
\pagebreak

$\fund{\gamma_2}{D_2}{\Rset^n}$
\pagebreak

$D = D_1 \times D_2$
\pagebreak

$\fund{\gamma}{D}{\Rset^n}$
\pagebreak

$d = (d_1, d_2) \in D$
\pagebreak

\[
  \gamma(d) = \gamma_1(d_1) \inters \gamma_2(d_2).
\]
\pagebreak

$d = (G, P) \in (\Gset \times \Pset)$
\pagebreak

$G$
\pagebreak

$[-\omega, \omega]$
\pagebreak

$\omega$
\pagebreak

$u \cdot 2^s$
\pagebreak

$u$
\pagebreak

$s$
\pagebreak

$\varepsilon_{\mathbf{f}}(l)$
\pagebreak

\[
\varepsilon_{\mathbf{f}}\left([a, b]+\sum_{v \in \cV}[a_{v}, b_{v}]v\right)
\defeq
(\textrm{max}(|a|, |b|) \amifp [-\beta^{-\textrm{p}}, \beta^{-\textrm{p}}])
+
\sum_{v \in \cV}(\textrm{max}(|a_{v}|,|b_{v}|)
\amifp
[-\beta^{-\textrm{p}}, \beta^{-\textrm{p}}])v
\]
\pagebreak

$\mathbf{f}$
\pagebreak

$\iota(lf)\rho^{\#}$
\pagebreak

\[
\iota\left(i + \sum_{v \in \cV}i_{v}v\right)\rho^{\#}
\defeq
i \asifp \left(\bigoplus_{v \in \cV}{}^{\#}i_{v} \amifp
\rho^{\#}(v)\right)
\]
\pagebreak

$\rho^{\#}(v)$
\pagebreak

$ y \cdot 2^\mathtt{exp} $
\pagebreak

$ y / 2^\mathtt{exp} $
\pagebreak

$15 \cdot 10^2 = 1500$
\pagebreak

$5$
\pagebreak

$3/580000$
\pagebreak

$713/10$
\pagebreak

$-1929/15625$
\pagebreak

$11/50$
\pagebreak

$-220001/100$
\pagebreak

$35$
\pagebreak

$44027$
\pagebreak

$7/2$
\pagebreak

$2000$
\pagebreak

$1024$
\pagebreak

$1088$
\pagebreak

$9073863231288$
\pagebreak

$17/8$
\pagebreak

$482$
\pagebreak

$256$
\pagebreak

\[
  \{\,
    a \in \mathtt{to}
  \mid
    \exists b \in \mathtt{x} \st a \mathrel{\mathtt{rel}} b
  \,\}.
\]
\pagebreak

\[
  \{\,
    a \in \mathtt{to}
  \mid
    \forall b \in \mathtt{x} \itc a \mathrel{\mathtt{rel}} b
  \,\}.
\]
\pagebreak

$ \{ x_1 \mapsto x_2, x_2 \mapsto x_3, x_3 \mapsto x_1 \}$
\pagebreak

$ x_1, x_2, x_3 $
\pagebreak

$O(n)$
\pagebreak

$O(1)$
\pagebreak

$O(n*\log^2 n)$
\pagebreak

$O(\log n)$
\pagebreak

$O(\log^2 n)$
\pagebreak

$O(k+\log^2 n)$
\pagebreak

$O(k+\log n)$
\pagebreak

$O(\log(last - first + 1))$
\pagebreak

$O(k)$
\pagebreak

$O(k*\log^2 n)$
\pagebreak

$O(k + \log m)$
\pagebreak

$0<=0$
\pagebreak

$0 \leq 1$
\pagebreak

$\sum_{i=0}^{n-1} 0 x_i + 0 = 0$
\pagebreak

$\sum_{i=0}^{n-1} 0 x_i + b \geq 0$
\pagebreak

$b \geq 0$
\pagebreak

$\sum_{i=0}^{n-1} 0 x_i + b > 0$
\pagebreak

$b > 0$
\pagebreak

$\sum_{i=0}^{n-1} 0 x_i + b = 0$
\pagebreak

$b \neq 0$
\pagebreak

$b < 0$
\pagebreak

$b \leq 0$
\pagebreak

$\epsilon \geq 0$
\pagebreak

$\epsilon \leq 1$
\pagebreak

$0<0$
\pagebreak

$L_\infty$
\pagebreak

$ e_1 = e_2
\pmod{m}$
\pagebreak

$ e_1 = e_2 \pmod{mk}$
\pagebreak

$\sum_{i=0}^{n-1} 0 x_i + 0 == 0$
\pagebreak

$\sum_{i=0}^{n-1} 0 x_i + b \%= 0 / m$
\pagebreak

$b = 0 \pmod{m}$
\pagebreak

$\sum_{i=0}^{n-1} 0 x_i + b == 0$
\pagebreak

$b \neq 0 \pmod{m}$
\pagebreak

$0 = 1 \pmod{1}$
\pagebreak

$0 = 1 \pmod{0}$
\pagebreak

$e1 = e2 \pmod{1}$
\pagebreak

$e = n \pmod{1}$
\pagebreak

$n = e \pmod{1}$
\pagebreak

$ e_1 = e_2
  \pmod{m}$
\pagebreak

$\mathrm{var}' \relsym \frac{\mathrm{expr}}{\mathrm{denominator}}$
\pagebreak

$\mathrm{lhs}' \relsym \mathrm{rhs}$
\pagebreak

$\frac{\mathrm{lb\_expr}}{\mathrm{denominator}}
     \leq \mathrm{var}'
       \leq \frac{\mathrm{ub\_expr}}{\mathrm{denominator}}$
\pagebreak

$\mathtt{y} \Delta \mathtt{x}$
\pagebreak

$\cB \sseq \Rset^2$
\pagebreak

\[
  \bigl\{\,
    (x, y, z)^\transpose \in \Rset^3
  \bigm|
    (x, y)^\transpose \in \cB
  \,\bigr\}.
\]
\pagebreak

\[
  \bigl\{\,
    (x, y, 0)^\transpose \in \Rset^3
  \bigm|
    (x, y)^\transpose \in \cB
  \,\bigr\}.
\]
\pagebreak

$B \sseq \Rset^n$
\pagebreak

$D \sseq \Rset^m$
\pagebreak

$R \sseq \Rset^{n+m}$
\pagebreak

\[
  R \defeq
      \Bigl\{\,
        (x_1, \ldots, x_n, y_1, \ldots, y_m)^\transpose
      \Bigm|
        (x_1, \ldots, x_n)^\transpose \in B,
        (y_1, \ldots, y_m)^\transpose \in D
      \,\Bigl\}.
\]
\pagebreak

$f(k)$
\pagebreak

$\dots$
\pagebreak

$I$
\pagebreak

$d$
\pagebreak

$n/d$
\pagebreak

$\times$
\pagebreak

$\epsilon$
\pagebreak

$M \in \Rset^r \times \Rset^c$
\pagebreak

$N \in \Rset^{r+n} \times \Rset^{c+n}$
\pagebreak

$N = \bigl(\genfrac{}{}{0pt}{}{0}{M}\genfrac{}{}{0pt}{}{J}{o}\bigr)$
\pagebreak

$n \times n$
\pagebreak

$\sum_{i = 0}^{n - 1} a_i x_i + b$
\pagebreak

${a'}_{ij}$
\pagebreak

$a_{ij}$
\pagebreak

\[
  {a'}_{ij} =
    \begin{cases}
      a_{ij} * \mathrm{denominator} + a_{iv} * \mathrm{expr}[j]
        \quad \text{for } j \neq v; \\
      \mathrm{expr}[v] * a_{iv}
        \quad \text{for } j = v.
    \end{cases}
\]
\pagebreak

$ e_1 = e_2 $
\pagebreak

$ e_1 = e_2
\pmod{0}$
\pagebreak

$r \times c$
\pagebreak

$(r+dims) \times
(c+dims)$
\pagebreak

$\bigl(\genfrac{}{}{0pt}{}{0}{A} \genfrac{}{}{0pt}{}{B}{A}\bigr)$
\pagebreak

$B$
\pagebreak

$dims \times dims$
\pagebreak

$\bigl(\genfrac{}{}{0pt}{}{0}{1} \genfrac{}{}{0pt}{}{1}{0}\bigr)$
\pagebreak

\[
  \frac{\sum_{i = 0}^{n - 1} a_i x_i + b}
       {\mathrm{denominator}}.
\]
\pagebreak

$\leq$
\pagebreak

$\relsym$
\pagebreak

\[
  \bigl\{\,
    (x, y, z)^\transpose \in \Rset^3
  \bigm|
    (x, y)^\transpose \in \cP
  \,\bigr\}.
\]
\pagebreak

\[
  \bigl\{\,
    (x, y, 0)^\transpose \in \Rset^3
  \bigm|
    (x, y)^\transpose \in \cP
  \,\bigr\}.
\]
\pagebreak

$\Rset^0 = \{\cdot\}$
\pagebreak

\[
  \begin{cases}
    sat\_c[i][j] = 0,
      \quad \text{if } G[i] \cdot C^\mathrm{T}[j] = 0; \\
    sat\_c[i][j] = 1,
      \quad \text{if } G[i] \cdot C^\mathrm{T}[j] > 0.
  \end{cases}
\]
\pagebreak

\[
  \begin{cases}
    sat\_g[i][j] = 0,
      \quad \text{if } C[i] \cdot G^\mathrm{T}[j] = 0; \\
    sat\_g[i][j] = 1,
      \quad \text{if } C[i] \cdot G^\mathrm{T}[j] > 0.
  \end{cases}
\]
\pagebreak

$O(r)$
\pagebreak

$O(n+k)$
\pagebreak

$O(\sum_{j=1}^{r} (k_j*\log^2 n_j))$
\pagebreak

$k_j$
\pagebreak

$n_j$
\pagebreak

$O(r+k*\log^2 c)$
\pagebreak

$(r+n) \times (c+m)$
\pagebreak

$\bigl(\genfrac{}{}{0pt}{}{M}{0} \genfrac{}{}{0pt}{}{0}{0}\bigr)$
\pagebreak

$(r+n) \times c$
\pagebreak

$\genfrac{(}{)}{0pt}{}{M}{0}$
\pagebreak

$(r+1) \times c$
\pagebreak

$\genfrac{(}{)}{0pt}{}{M}{y}$
\pagebreak

$ \{ 1 \mapsto 3, 2 \mapsto 4,
3 \mapsto 6, 4 \mapsto 2, 5 \mapsto 5, 6 \mapsto 1 \}$
\pagebreak

$(1 3 6)(2 4)$
\pagebreak

$O(k*\sum_{j=1}^{r} \log^2 n_j)$
\pagebreak

$O(k*r*\log^2 c)$
\pagebreak

$r \times (c+n)$
\pagebreak

$(M \, 0)$
\pagebreak

$O(\sum_{j=1}^{r} (k_j+\log n_j))$
\pagebreak

$O(k+r*\log c)$
\pagebreak

$O(k + \sum_{j=1}^{r} (\log^2 n_j))$
\pagebreak

$O(r*(c-i+\log^2 c))$
\pagebreak

$O(\sum_{j=1}^r (k_j*\log n_j))$
\pagebreak

$O(r*n*\log c)$
\pagebreak

$O(n*\log n)$
\pagebreak

$O(r+k)$
\pagebreak

$[\mathrm{first}, \mathrm{last})$
\pagebreak

$\frac{\mathtt{numer}}{\mathtt{denom}}$
\pagebreak

$\frac{d_{j}}{\|\Delta x^{j} \|}$
\pagebreak

\[
  \|\Delta x^{j} \|
    = \left(
        1+\sum_{i=1}^{m} \alpha_{ij}^2
      \right)^{\frac{1}{2}}.
\]
\pagebreak

$\alpha$
\pagebreak

$\rho^{\#}$
\pagebreak

$\rho^{\#}_l$
\pagebreak

\[
  \langle \vect{c}, \vect{r}_1 \rangle = 0 \land
  \langle \vect{c}, \vect{r}_2 \rangle = 0
  \Rightarrow
  \langle \vect{c}, (\lambda_1 \vect{r}_1 + \lambda_2 \vect{r}_2) \rangle =
  \lambda_1 \langle \vect{c}, \vect{r}_1 \rangle
  + \lambda_2 \langle \vect{c}, \vect{r}_2 \rangle
  = 0,
\]
\pagebreak

$\lambda_1, \lambda_2$
\pagebreak

$(r+dims)
\times (c+dims)$
\pagebreak

$\bigl(\genfrac{}{}{0pt}{}{A}{0} \genfrac{}{}{0pt}{}{0}{B}\bigr)$
\pagebreak

$\bigl(\genfrac{}{}{0pt}{}{1}{0} \genfrac{}{}{0pt}{}{0}{1}\bigr)$
\pagebreak

$\mathrm{var}' = \frac{\mathrm{expr}}{\mathrm{denominator}}
\pmod{\mathrm{modulus}}$
\pagebreak

$\mathrm{lhs}' = \mathrm{rhs} \pmod{\mathrm{modulus}}$
\pagebreak

$\cL \sseq
\Rset^2$
\pagebreak

\[
  \bigl\{\,
    (x, y, z)^\transpose \in \Rset^3
  \bigm|
    (x, y)^\transpose \in \cL
  \,\bigr\}.
\]
\pagebreak

$\cL \sseq \Rset^2$
\pagebreak

\[
  \bigl\{\,
    (x, y, 0)^\transpose \in \Rset^3
  \bigm|
    (x, y)^\transpose \in \cL
  \,\bigr\}.
\]
\pagebreak

$\{3x \equiv_3 0, 4x + y \equiv_3 1\}$
\pagebreak

$\{3x \equiv_3 0, x + y \equiv_3 1\}$
\pagebreak

$\mathrm{var} \relsym \frac{\mathrm{expr}}{\mathrm{denominator}}$
\pagebreak

$l = c$
\pagebreak

$l = ax + c$
\pagebreak

$l = ax + by + c$
\pagebreak

$ax + b = cy + d$
\pagebreak

$\frac{\mathrm{lb\_expr}}{\mathrm{denominator}}
     \leq \mathrm{var}'
     \leq \frac{\mathrm{ub\_expr}}{\mathrm{denominator}}$
\pagebreak

$\cO \sseq \Rset^2$
\pagebreak

\[
  \bigl\{\,
    (x, y, z)^\transpose \in \Rset^3
  \bigm|
    (x, y)^\transpose \in \cO
  \,\bigr\}.
\]
\pagebreak

\[
  \bigl\{\,
    (x, y, 0)^\transpose \in \Rset^3
  \bigm|
    (x, y)^\transpose \in \cO
  \,\bigr\}.
\]
\pagebreak

$O(n^2)$
\pagebreak

$i + \sum_{v \in \cV}i_{v}v $
\pagebreak

$i' + \sum_{v \in \cV}i'_{v}v $
\pagebreak

$\aslf$
\pagebreak

\[
\left(i + \sum_{v \in \cV}i_{v}v \right)
\aslf
\left(i' + \sum_{v \in \cV}i'_{v}v \right)
=
\left(i \asifp i'\right)
+ \sum_{v \in \cV}\left(i_{v} \asifp i'_{v} \right)v.
\]
\pagebreak

$e_{1} \oplus e_{2}$
\pagebreak

$\left \llbracket \rho^{\#}, \rho^{\#}_l \right
\rrbracket$
\pagebreak

$\linexprenv{e_{1} \oplus e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\pagebreak

\[
\linexprenv{e_{1} \oplus e_{2}}{\rho^{\#}}{\rho^{\#}_l}
=
\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l}
\aslf
\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l}
\aslf
\varepsilon_{\mathbf{f}}\left(\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l}
\right)
\aslf
\varepsilon_{\mathbf{f}}\left(\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l}
\right)
\aslf
mf_{\mathbf{f}}[-1, 1]
\]
\pagebreak

$l$
\pagebreak

$mf_{\mathbf{f}}$
\pagebreak

$\adlf$
\pagebreak

\[
\left(i + \sum_{v \in \cV}i_{v}v\right)
\aslf
\left(i' + \sum_{v \in \cV}i'_{v}v\right)
=
\left(i \asifp i'\right)
+ \sum_{v \in \cV}\left(i_{v} \asifp i'_{v}\right)v,
\]
\pagebreak

\[
\left(i + \sum_{v \in \cV}i_{v}v\right)
\adlf
\left(i' + \sum_{v \in \cV}i'_{v}v\right)
=
\left(i \adifp i'\right)
+ \sum_{v \in \cV}\left(i_{v} \adifp i'_{v}\right)v.
\]
\pagebreak

$e_{1} \ominus e_{2}$
\pagebreak

$\linexprenv{e_{1} \ominus e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\pagebreak

\[
\linexprenv{e_{1} \ominus e_{2}}{\rho^{\#}}{\rho^{\#}_l}
=
\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l}
\adlf
\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l}
\aslf
\varepsilon_{\mathbf{f}}\left(\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l}
\right)
\aslf
\varepsilon_{\mathbf{f}}\left(\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l}
\right)
\aslf
mf_{\mathbf{f}}[-1, 1]
\]
\pagebreak

$\amlf$
\pagebreak

\[
i
\amlf
\left(i' + \sum_{v \in \cV}i'_{v}v\right)
=
\left(i \amifp i'\right)
+ \sum_{v \in \cV}\left(i \amifp i'_{v}\right)v.
\]
\pagebreak

$[a, b] \otimes e_{2}$
\pagebreak

$\linexprenv{[a, b] \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\pagebreak

\[
\linexprenv{[a, b] \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l}
=
\left([a, b]
\amlf
\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l}\right)
\aslf
\left([a, b]
\amlf
\varepsilon_{\mathbf{f}}\left(\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l}
\right)\right)
\aslf
mf_{\mathbf{f}}[-1, 1].
\]
\pagebreak

$e_{1} \otimes [a, b]$
\pagebreak

$\linexprenv{e_{1} \otimes [a, b]}{\rho^{\#}}{\rho^{\#}_l}$
\pagebreak

\[
\linexprenv{e_{1} \otimes [a, b]}{\rho^{\#}}{\rho^{\#}_l}
=
\linexprenv{[a, b] \otimes e_{1}}{\rho^{\#}}{\rho^{\#}_l}.
\]
\pagebreak

$e_{1} \otimes e_{2}$
\pagebreak

$\linexprenv{e_{1} \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\pagebreak

\[
\linexprenv{e_{1} \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l}
=
\linexprenv{\iota\left(\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l}
\right)\rho^{\#}
\otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l},
\]
\pagebreak

$\iota(l)\rho^{\#}$
\pagebreak

$\adivlf$
\pagebreak

\[
\left(i + \sum_{v \in \cV}i_{v}v\right)
\adivlf
i'
=
\left(i \adivifp i'\right)
+ \sum_{v \in \cV}\left(i_{v} \adivifp i'\right)v.
\]
\pagebreak

$e_{1} \oslash [a, b]$
\pagebreak

$
\linexprenv{e_{1} \oslash [a, b]}{\rho^{\#}}{\rho^{\#}_l}
$
\pagebreak

\[
\linexprenv{e_{1} \oslash [a, b]}{\rho^{\#}}{\rho^{\#}_l}
=
\left(\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l}
\adivlf
[a, b]\right)
\aslf
\left(\varepsilon_{\mathbf{f}}\left(
\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l}
\right)
\adivlf
[a, b]\right)
\aslf
mf_{\mathbf{f}}[-1, 1],
\]
\pagebreak

$e_{1} \oslash e_{2}$
\pagebreak

$\linexprenv{e_{1} \oslash e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\pagebreak

\[
\linexprenv{e_{1} \oslash e_{2}}{\rho^{\#}}{\rho^{\#}_l}
=
\linexprenv{e_{1} \oslash \iota\left(
\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l}
\right)\rho^{\#}}{\rho^{\#}}{\rho^{\#}_l},
\]
\pagebreak

$\frac{\mathtt{expr}}{\mathtt{d}}$
\pagebreak

$f_i(x,p) = 0 ; 1 \leq i \leq n$
\pagebreak

$f_i(x,p) \geq 0 ; 1 \leq i \leq n$
\pagebreak

$\sum\limits_{i=1}^n f_i(x,p) \leq 0$
\pagebreak

$c=s_{*j}\frac{t_{ik}}{s_{ij}}$
\pagebreak

$c'=s_{*j'}\frac{t_{i'k}}{s_{i'j'}}$
\pagebreak

$s_{*j}$
\pagebreak

$s_{*j'}$
\pagebreak

$j'$
\pagebreak

$(i,j)$
\pagebreak

$c'$
\pagebreak

$(i',j')$
\pagebreak

$-c$
\pagebreak

$-c'$
\pagebreak

\[
  \mathit{ps} = \bigl\{ \{ x \geq 0 \}, \{ x \leq 0 \} \bigr\},
\]
\pagebreak

$\mathit{ps}$
\pagebreak

$\cP \uplus \cQ \neq \cP \union \cQ$
\pagebreak

$ x'_1, \ldots, x'_n $
\pagebreak

$ 0, \ldots, n-1 $
\pagebreak

$ x_1, \ldots, x_n $
\pagebreak

$ n, \ldots, 2n-1 $
\pagebreak

$ n+1 $
\pagebreak

$ \mu_0 + \sum_{i=1}^n \mu_i x_i $
\pagebreak

$ \mu_0, \mu_1, \ldots, \mu_n $
\pagebreak

$ n, 0, \ldots, n-1 $
\pagebreak

$\linexprenv{cast(e)}{\rho^{\#}}{\rho^{\#}_l}$
\pagebreak

\[
\linexprenv{cast(e)}{\rho^{\#}}{\rho^{\#}_l}
=
\linexprenv{e}{\rho^{\#}}{\rho^{\#}_l}
\aslf
\varepsilon_{\mathbf{f}}\left(\linexprenv{e}{\rho^{\#}}{\rho^{\#}_l}
\right)
\aslf
mf_{\mathbf{f}}[-1, 1]
\]
\pagebreak

$[l, u]$
\pagebreak

$\linexprenv{v}{\rho^{\#}}{\rho^{\#}_l}$
\pagebreak

$\rho^{\#}_l(v)$
\pagebreak

$[-1, 1]v$
\pagebreak

$\oplus$
\pagebreak

$\ominus$
\pagebreak

$\otimes$
\pagebreak

$\oslash$
\pagebreak

\[
\adlf
\left(i + \sum_{v \in \cV}i_{v}v\right)
=
\left(\adifp i\right)
+ \sum_{v \in \cV}\left(\adifp i_{v} \right)v,
\]
\pagebreak

$\ominus e$
\pagebreak

$\linexprenv{\ominus e}{\rho^{\#}}{\rho^{\#}_l}$
\pagebreak

\[
\linexprenv{\ominus e}{\rho^{\#}}{\rho^{\#}_l}
=
\adlf
\left(
\linexprenv{e}{\rho^{\#}}{\rho^{\#}_l}
\right).
\]
\pagebreak

\end{document}
