\form#0:$n$
\form#1:$ \Rset^n $
\form#2:$0 \geq 2$
\form#3:$0 \leq 2$
\form#4:$x \leq 2$
\form#5:$x - y \leq 2$
\form#6:$x + y \leq 2$
\form#7:$x = 2$
\form#8:$x - 3y \leq 2$
\form#9:$x - 3y < 2$
\form#10:$0 \equiv_2 1$
\form#11:$0 \equiv_2 2$
\form#12:$x + 3y \equiv_0 0$
\form#13:$x + 3y \equiv_5 0$
\form#14:$2x \leq 5$
\form#15:$x \leq 3$
\form#16:$w\textrm{-bit}$
\form#17:$2^w$
\form#18:$x \bmod 2^w$
\form#19:\[ \mathrm{wrap}^\mathrm{u}_w(x) \defeq x - 2^w \lfloor x/2^w \rfloor. \]
\form#20:\[ \mathrm{wrap}^\mathrm{s}_w(x) \defeq \begin{cases} \mathrm{wrap}^\mathrm{u}_w(x), &\text{if $\mathrm{wrap}^\mathrm{u}_w(x) < 2^{w-1}$;} \\ \mathrm{wrap}^\mathrm{u}_w(x) - 2^w, &\text{otherwise.} \end{cases} \]
\form#21:$x$
\form#22:$S \sseq \Rset$
\form#23:$\bigl\{\, \mathrm{wrap}^\mathrm{u}_w(z) \bigm| z \in S \,\bigr\}$
\form#24:$\Rset^n$
\form#25:$n\textrm{-dimensional}$
\form#26:$\Rset$
\form#27:$\nonnegRset$
\form#28:$i \in \{0, \ldots, n-1\}$
\form#29:$v_i$
\form#30:$i\textrm{-th}$
\form#31:$\vect{v} = (v_0, \ldots, v_{n-1})^\transpose \in \Rset^n$
\form#32:$\vect{0}$
\form#33:$\vect{v} \in \Rset^n$
\form#34:$\Rset^{n \times 1}$
\form#35:$\vect{v}^\transpose$
\form#36:$\vect{v},\vect{w} \in \Rset^n$
\form#37:$\langle \vect{v}, \vect{w} \rangle$
\form#38:\[ \vect{v}^\transpose \vect{w} = \sum_{i=0}^{n-1} v_i w_i. \]
\form#39:$S_1, S_2 \sseq \Rset^n$
\form#40:$S_1$
\form#41:$S_2$
\form#42:$S_1 + S_2 = \{\, \vect{v}_1 + \vect{v}_2 \mid \vect{v}_1 \in S_1, \vect{v}_2 \in S_2 \,\}.$
\form#43:$\vect{a} \in \Rset^n$
\form#44:$b \in \Rset$
\form#45:$\vect{a} \neq \vect{0}$
\form#46:$\mathord{\relsym} \in \{ =, \geq, > \}$
\form#47:$\langle \vect{a}, \vect{x} \rangle \relsym b$
\form#48:$\mathord{\relsym} \in \{ = \}$
\form#49:$\mathord{\relsym} \in \{ \geq \}$
\form#50:$\mathord{\relsym} \in \{ > \}$
\form#51:$\langle \vect{a}, \vect{x} \rangle = b$
\form#52:$\langle \vect{a}, \vect{x} \rangle \geq b$
\form#53:$\langle -\vect{a}, \vect{x} \rangle \geq -b$
\form#54:$\vect{a} = \vect{0}$
\form#55:$\langle \vect{0}, \vect{x} \rangle \relsym b$
\form#56:$\emptyset$
\form#57:$\cP \sseq \Rset^n$
\form#58:$\cP$
\form#59:$n = 0$
\form#60:$\cP = \emptyset$
\form#61:$\Pset_n$
\form#62:$\cP \in \Pset_n$
\form#63:$\CPset_n$
\form#64:$\lambda \in \nonnegRset$
\form#65:\[ \cP \sseq \bigl\{\, \vect{x} \in \Rset^n \bigm| - \lambda \leq x_j \leq \lambda \text{ for } j = 0, \ldots, n-1 \,\bigr\}. \]
\form#66:\[ \cP \defeq \{\, \vect{x} \in \Rset^n \mid A_1 \vect{x} = \vect{b}_1, A_2 \vect{x} \geq \vect{b}_2, A_3 \vect{x} > \vect{b}_3 \,\}, \]
\form#67:$i \in \{1, 2, 3\}$
\form#68:$A_i \in \Rset^{m_i} \times \Rset^n$
\form#69:$\vect{b}_i \in \Rset^{m_i}$
\form#70:$m_1, m_2, m_3 \in \Nset$
\form#71:$S = \{ \vect{x}_1, \ldots, \vect{x}_k \} \sseq \Rset^n$
\form#72:$\lambda_1, \ldots, \lambda_k \in \Rset$
\form#73:$\vect{v} = \sum_{j=1}^k \lambda_j \vect{x}_j$
\form#74:$S$
\form#75:$\forall j \in \{ 1, \ldots, k \} \itc \lambda_j \in \nonnegRset$
\form#76:$\sum_{j = 1}^k \lambda_j = 1$
\form#77:$\linearhull(S)$
\form#78:$\conichull(S)$
\form#79:$\affinehull(S)$
\form#80:$\convexhull(S)$
\form#81:$P, C \sseq \Rset^n$
\form#82:$P \union C = S$
\form#83:$\NNChull(P, C)$
\form#84:$\lambda_j > 0$
\form#85:$\vect{x}_j \in P$
\form#86:$P$
\form#87:$\NNChull(P, C) = \NNChull(P, P \union C)$
\form#88:$C \sseq P$
\form#89:\[ \convexhull(P) = \NNChull(P, \emptyset) = \NNChull(P, P) = \NNChull(P, C). \]
\form#90:$\vect{p} \in \cP$
\form#91:$\vect{c} \in \Rset^n$
\form#92:$\vect{r} \in \Rset^n$
\form#93:$\vect{r} \neq \vect{0}$
\form#94:$\cP \neq \emptyset$
\form#95:$\vect{p} + \lambda \vect{r} \in \cP$
\form#96:$\vect{l} \in \Rset^n$
\form#97:$\vect{l}$
\form#98:$-\vect{l}$
\form#99:$\vect{r}$
\form#100:$\vect{r}_1$
\form#101:$\vect{r}_2$
\form#102:$\vect{r} \neq \lambda \vect{r}_1$
\form#103:$\vect{r} \neq \lambda \vect{r}_2$
\form#104:$\vect{r}_1 \neq \lambda \vect{r}_2$
\form#105:$L$
\form#106:$R$
\form#107:$C$
\form#108:$\cG = (L, R, P, C)$
\form#109:\[ \cP = \linearhull(L) + \conichull(R) + \NNChull(P, C), \]
\form#110:$+$
\form#111:$\cP \in \CPset_n$
\form#112:$\cG = (L, R, P)$
\form#113:\[ \cP = \linearhull(L) + \conichull(R) + \convexhull(P). \]
\form#114:$P = \emptyset$
\form#115:$\Rset^2$
\form#116:$y \geq 0$
\form#117:$L = \bigl\{ (1, 0)^\transpose \bigr\}$
\form#118:$R = \bigl\{ (0, 1)^\transpose \bigr\}$
\form#119:$P = \bigl\{ (0, 0)^\transpose \bigr\}$
\form#120:$C = \emptyset$
\form#121:$\cC$
\form#122:$\cG' = (L', R', P', C') \neq \cG$
\form#123:$L' \sseq L$
\form#124:$R' \sseq R$
\form#125:$P' \sseq P$
\form#126:$C' \sseq C$
\form#127:$\cG$
\form#128:$(\cC, \cG)$
\form#129:$(L, R, P)$
\form#130:$n \in \Nset$
\form#131:$\vect{a}, \vect{x} \in \Rset^m$
\form#132:$m \leq n$
\form#133:$\vect{x} \in \Rset^m$
\form#134:$\{ \vect{x}_1, \ldots, \vect{x}_k \} \sseq \Rset^n$
\form#135:\[ \sum_{i = 1}^k \lambda_i \vect{x}_i = \vect{0}, \quad \sum_{i = 1}^k \lambda_i = 0 \]
\form#136:$i = 1, \ldots, k$
\form#137:$\lambda_i = 0$
\form#138:$n + 1$
\form#139:$k \in \Nset$
\form#140:$\pdim(\cP) = k$
\form#141:$k + 1$
\form#142:$0 \leq \pdim(\cP) \leq n$
\form#143:$-1$
\form#144:$k \leq n$
\form#145:$\pdim(\cP) \neq \pdim(\cQ)$
\form#146:$\cQ$
\form#147:$\pdim(\cP) = \pdim(\cQ)$
\form#148:$\cP_1, \cP_2 \in \Pset_n$
\form#149:$\cP_1$
\form#150:$\cP_2$
\form#151:$\cP_1 \inters \cP_2$
\form#152:$\cP_1 \uplus \cP_2$
\form#153:$\cP_1, \cP_2 \in \CPset_n$
\form#154:$\cQ \in \Pset_m$
\form#155:$\cR \in \Pset_{n+m}$
\form#156:\[ \cR \defeq \Bigl\{\, (x_0, \ldots, x_{n-1}, y_0, \ldots, y_{m-1})^\transpose \in \Rset^{n+m} \Bigm| (x_0, \ldots, x_{n-1})^\transpose \in \cP, (y_0, \ldots, y_{m-1})^\transpose \in \cQ \,\Bigl\}. \]
\form#157:$n+m$
\form#158:$i$
\form#159:$\cQ \in \Pset_{n+i}$
\form#160:$i+n$
\form#161:$\cP \sseq \Rset^2$
\form#162:\[ \cQ = \bigl\{\, (x_0, x_1, x_2)^\transpose \in \Rset^3 \bigm| (x_0, x_1)^\transpose \in \cP \,\bigr\}. \]
\form#163:\[ \cQ = \bigl\{\, (x_0, x_1, 0)^\transpose \in \Rset^3 \bigm| (x_0, x_1)^\transpose \in \cP \,\bigr\}. \]
\form#164:$\cP \in \Pset_4$
\form#165:$\bigl\{ (3, 1, 0, 2)^\transpose \bigr\} \sseq \Rset^4$
\form#166:$\{x_1, x_2\}$
\form#167:\[ \cQ = \bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2. \]
\form#168:$m$
\form#169:$m = 2$
\form#170:\[ \cQ = \bigl\{ (3, 1)^\transpose \bigr\} \sseq \Rset^2. \]
\form#171:$\pard{\rho}{\{0, \ldots, n-1\}}{\Nset}$
\form#172:$\rho\bigl(\{0, \ldots, n-1\}\bigr) = \{0, \ldots, m-1\}$
\form#173:$\rho$
\form#174:$m = 0$
\form#175:$\Rset^0$
\form#176:\[ \cQ \defeq \Bigl\{\, \bigl(v_{\rho^{-1}(0)}, \ldots, v_{\rho^{-1}(m-1)}\bigr)^\transpose \Bigm| (v_0, \ldots, v_{n-1})^\transpose \in \cP \,\Bigr\}. \]
\form#177:$n > 0$
\form#178:$n+1$
\form#179:$\ldots$
\form#180:$n+m-1$
\form#181:\[ \cQ \defeq \sset{ \vect{u} \in \Rset^{n+m} }{ \exists \vect{v}, \vect{w} \in \cP \st u_i = v_i \\ \qquad \mathord{} \land \forall j = n, n+1, \ldots, n+m-1 \itc u_j = w_i \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_k = v_k = w_k }. \]
\form#182:$J = \{ j_0, \ldots, j_{m-1} \}$
\form#183:$m < n$
\form#184:$j < n$
\form#185:$j \in J$
\form#186:$i < n$
\form#187:$i \notin J$
\form#188:\[ \cQ \defeq \biguplus_{d = 0}^m \cQ_d \]
\form#189:\[ \cQ_m \defeq \sset{ \vect{u} \in \Rset^{n-m} }{ \exists \vect{v} \in \cP \st u_{i'} = v_i \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_{k'} = v_k } \]
\form#190:$ d = 0 $
\form#191:$ \ldots $
\form#192:$ m-1 $
\form#193:\[ \cQ_d \defeq \sset{ \vect{u} \in \Rset^{n-m} }{ \exists \vect{v} \in \cP \st u_{i'} = v_{j_d} \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_{k'} = v_k }, \]
\form#194:$ k = 0 $
\form#195:$ n-1 $
\form#196:\[ k' \defeq k - \card \{\, j \in J \mid k > j \,\}, \]
\form#197:$\card S$
\form#198:$\reld{\phi}{\Rset^n}{\Rset^m}$
\form#199:$\phi(S) \sseq \Rset^m$
\form#200:$\phi$
\form#201:$S \sseq \Rset^n$
\form#202:\[ \phi(S) \defeq \bigl\{\, \vect{w} \in \Rset^m \bigm| \exists \vect{v} \in S \st (\vect{v}, \vect{w}) \in \phi \,\bigr\}. \]
\form#203:$\phi^{-1}(S') \sseq \Rset^n$
\form#204:$S' \sseq \Rset^m$
\form#205:\[ \phi^{-1}(S') \defeq \bigl\{\, \vect{v} \in \Rset^n \bigm| \exists \vect{w} \in S' \st (\vect{v}, \vect{w}) \in \phi \,\bigr\}. \]
\form#206:$n = m$
\form#207:$\ell \in \Nset$
\form#208:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^m \itc (\vect{v}, \vect{w}) \in \phi \iff \bigland_{i=1}^{\ell} \bigl( \langle \vect{c}_i, \vect{w} \rangle \relsym_i \langle \vect{a}_i, \vect{v} \rangle + b_i \bigr), \]
\form#209:$\vect{a}_i \in \Rset^n$
\form#210:$\vect{c}_i \in \Rset^m$
\form#211:$b_i \in \Rset$
\form#212:$\mathord{\relsym}_i \in \{ <, \leq, =, \geq, > \}$
\form#213:$i = 1, \ldots, \ell$
\form#214:$A \in \Rset^m \times \Rset^n$
\form#215:$\vect{b} \in \Rset^m$
\form#216:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^m \itc (\vect{v}, \vect{w}) \in \phi \iff \vect{w} = A\vect{v} + \vect{b}. \]
\form#217:$<$
\form#218:$>$
\form#219:$\reld{\phi}{\Rset^n}{\Rset^n}$
\form#220:$\vect{x} = (x_0, \ldots, x_{n-1})^\transpose$
\form#221:$\vect{x}' = (x'_0, \ldots, x'_{n-1})^\transpose$
\form#222:$x'_i$
\form#223:$x'_i = x_i$
\form#224:$\reld{\phi}{\Rset^3}{\Rset^3}$
\form#225:$x'_0 - x'_2 \geq 2 x_0 - x_1$
\form#226:$x'_1$
\form#227:\[ \forall \vect{v} \in \Rset^3, \vect{w} \in \Rset^3 \itc (\vect{v}, \vect{w}) \in \phi \iff (w_0 - w_2 \geq 2 v_0 - v_1) \land (w_1 = v_1). \]
\form#228:$x'_0 + 0 \cdot x'_1 - x'_2 \geq 2 x_0 - x_1$
\form#229:$x'_k$
\form#230:$\langle \vect{a}, \vect{x} \rangle + b$
\form#231:$\fund{\phi = \bigl(x'_k = \langle \vect{a}, \vect{x} \rangle + b\bigr)} {\Rset^n}{\Rset^n}$
\form#232:\[ \forall \vect{v} \in \Rset^n \itc \phi(\vect{v}) = A\vect{v} + \vect{b}, \]
\form#233:\[ A = \begin{pmatrix} 1 & & 0 & 0 & \cdots & \cdots & 0 \\ & \ddots & & \vdots & & & \vdots \\ 0 & & 1 & 0 & \cdots & \cdots & 0 \\ a_0 & \cdots & a_{k-1} & a_k & a_{k+1} & \cdots & a_{n-1} \\ 0 & \cdots & \cdots & 0 & 1 & & 0 \\ \vdots & & & \vdots & & \ddots & \\ 0 & \cdots & \cdots & 0 & 0 & & 1 \end{pmatrix}, \qquad \vect{b} = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ b \\ 0 \\ \vdots \\ 0 \end{pmatrix} \]
\form#234:$a_i$
\form#235:$b$
\form#236:$(k+1)$
\form#237:$A$
\form#238:$\vect{b}$
\form#239:$(v_0, \ldots, v_{n-1})^\transpose$
\form#240:\[ \Bigl(v_0, \ldots, \bigl(\textstyle{\sum_{i=0}^{n-1}} a_i v_i + b\bigr), \ldots, v_{n-1}\Bigr)^\transpose. \]
\form#241:$x'_k = \langle \vect{a}, \vect{x} \rangle + b$
\form#242:$\bigl\{ (0, 0)^\transpose, (0, 3)^\transpose, (3, 0)^\transpose, (3, 3)^\transpose \bigr\}$
\form#243:$x_0$
\form#244:$x_0 + 2 x_1 + 4$
\form#245:$k = 0$
\form#246:$a_0 = 1, a_1 = 2, b = 4$
\form#247:$\bigl\{ (4, 0)^\transpose, (10, 3)^\transpose, (7, 0)^\transpose, (13, 3)^\transpose \bigr\}$
\form#248:$x_0 - 2 x_1$
\form#249:$x_1$
\form#250:$a_0 = 0, a_1 = 1, b = 0$
\form#251:$(1, 0)^\transpose$
\form#252:$a_k$
\form#253:$\mathrm{lb} = \langle \vect{a}, \vect{x} \rangle + b$
\form#254:$\mathrm{ub} = \langle \vect{c}, \vect{x} \rangle + d$
\form#255:$\phi = (\mathrm{lb} \leq x'_k \leq \mathrm{ub})$
\form#256:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n \itc (\vect{v}, \vect{w}) \in \phi \iff \bigl( \langle \vect{a}, \vect{v} \rangle + b \leq w_k \leq \langle \vect{c}, \vect{v} \rangle + d \bigr) \land \Bigl( \bigland_{0 \leq i < n, i \neq k} w_i = v_i \Bigr). \]
\form#257:$\mathbb{F}_{\mathrm{f}}$
\form#258:$\mathrm{f}$
\form#259:$\mathbb{I}_\mathrm{f}$
\form#260:$\langle\vect{\alpha}, \vect{x}\rangle + \beta$
\form#261:\[\left<\vect{\alpha}, \vect{x}\right> + \beta = \alpha_0x_0 + \ldots + \alpha_{n-1}x_{n-1} + \beta,\]
\form#262:$\beta$
\form#263:$\vect{\alpha}_k \in \mathbb{I}_{\mathrm{f}}$
\form#264:$k = 0, \ldots, n - 1$
\form#265:$\mathrm{lf}$
\form#266:$\mathrm{lb} \leq x'_k \leq \mathrm{ub}$
\form#267:$\mathrm{lb}$
\form#268:$\mathrm{ub}$
\form#269:$\phi = (\mathrm{lhs}' \relsym \mathrm{rhs})$
\form#270:$\mathrm{lhs} = \langle \vect{c}, \vect{x} \rangle + d$
\form#271:$\mathrm{rhs} = \langle \vect{a}, \vect{x} \rangle + b$
\form#272:$\mathord{\relsym} \in \{ <, \leq, =, \geq, > \}$
\form#273:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n \itc (\vect{v}, \vect{w}) \in \phi \iff \bigl( \langle \vect{c}, \vect{w} \rangle + d \relsym \langle \vect{a}, \vect{v} \rangle + b \bigr) \land \Bigl( \bigland_{0 \leq i < n, c_i = 0} w_i = v_i \Bigr). \]
\form#274:$\mathrm{lhs} = x_k$
\form#275:$x'_k = \mathrm{rhs}$
\form#276:$\mathrm{lhs}$
\form#277:$\mathrm{rhs}$
\form#278:$\mathrm{lhs}' \leq \mathrm{rhs}$
\form#279:$\mathrm{rhs}' \geq \mathrm{lhs}$
\form#280:$\cQ \in \Pset_n$
\form#281:$i \in \{ 0, \ldots, n-1 \}$
\form#282:\[ \cQ = \bigl\{\, \vect{w} \in \Rset^n \bigm| \exists \vect{v} \in \cP \st \forall j \in \{0, \ldots, n-1\} \itc j \neq i \implies w_j = v_j \,\bigr\}. \]
\form#283:$\cP, \cQ \in \Pset_n$
\form#284:$ \cP \nearrow \cQ$
\form#285:\[ \bigl\{\, \vect{p} + \lambda \vect{q} \in \Rset^n \bigm| \vect{p} \in \cP, \vect{q} \in \cQ, \lambda \in \nonnegRset \,\bigr\}. \]
\form#286:$\cP,\cQ \in \CPset_n$
\form#287:$\cP, \cQ, \cR \in \Pset_n$
\form#288:$\cR$
\form#289:$\cR \inters \cQ = \cP \inters \cQ$
\form#290:$\cR \Sseq \cP$
\form#291:$r \leq p$
\form#292:$r$
\form#293:$p$
\form#294:$ c = \bigl( \langle \vect{a}, \vect{x} \rangle \relsym b \bigr) $
\form#295:$c$
\form#296:$\cP \inters \cQ = \emptyset$
\form#297:$\cP \inters \cQ \neq \emptyset$
\form#298:$\cP \inters \cQ \subset \cP$
\form#299:$\cP \sseq \cQ$
\form#300:$\cP \sseq \cH$
\form#301:$\cH$
\form#302:$g$
\form#303:$\cP \widen \cQ$
\form#304:$\cP, \cQ \in \CPset_n$
\form#305:$\cP \Sseq \cP \widen \cQ$
\form#306:$\cQ \sseq \cP$
\form#307:$\cP = \cQ$
\form#308:$k$
\form#309:$\mathord{\relsym} \in \{ =, \geq, >\}$
\form#310:$k \in \{ 0, \ldots, i-1, i+1, \ldots, n-1 \}$
\form#311:$a_k = 0$
\form#312:$x = r$
\form#313:$x \leq r$
\form#314:$x \geq r$
\form#315:$x < r$
\form#316:$x > r$
\form#317:$r \in \Rset$
\form#318:$\cB$
\form#319:$\vect{e}_i = (0, \ldots, 1, \ldots, 0)^\transpose$
\form#320:$\langle \vect{e}_i, \vect{x} \rangle \relsym b$
\form#321:$\mathord{\relsym}$
\form#322:$\mathord{\geq}$
\form#323:$\mathord{>}$
\form#324:$\langle\vect{e}_i,\vect{x}\rangle \relsym b$
\form#325:$\mathord{\leq}$
\form#326:$\mathord{<}$
\form#327:$\{-2, -1, 0, 1, 2\}$
\form#328:$\mathord{\relsym} \in \{ =, \geq\}$
\form#329:$i, j \in \{ 0, \ldots, n-1 \}$
\form#330:$a_i, a_j \in \{ -1, 0, 1 \}$
\form#331:$a_i \neq a_j$
\form#332:$k \notin \{ i, j \}$
\form#333:$a, b, f \in \Rset$
\form#334:$a \equiv_f b$
\form#335:$\exists \mu \in \Zset \st a - b = \mu f$
\form#336:$\Sset \in \{ \Qset, \Rset \}$
\form#337:$\vect{a} \in \Sset^n \setdiff \{\vect{0}\}$
\form#338:$b, f \in \Sset$
\form#339:$\langle \vect{a}, \vect{x} \rangle \equiv_f b$
\form#340:$\Sset^n$
\form#341:\[ \bigl\{\, \vect{v} \in \Rset^n \bigm| \exists \mu \in \Zset \st \langle \vect{a}, \vect{v} \rangle = b + \mu f \,\bigr\}; \]
\form#342:$f \neq 0$
\form#343:$\langle \vect{a}, \vect{x} \rangle \equiv_0 b$
\form#344:$f = 0$
\form#345:$f$
\form#346:\[ \big\{\, \bigl(\langle \vect{a}, \vect{x} \rangle = b + \mu f\bigr) \bigm| \mu \in \Zset \,\bigr\}; \]
\form#347:$b \equiv_f 0$
\form#348:$\langle \vect{0}, \vect{x} \rangle \equiv_f b$
\form#349:$\cL \sseq \Rset^n$
\form#350:$\cL$
\form#351:$\Qset^n$
\form#352:$\cL = \emptyset$
\form#353:$\Gset_{n}$
\form#354:$\bigl\{\langle\vect{0}, \vect{x}\rangle \equiv_0 1\bigr\}$
\form#355:$0 = 1$
\form#356:$\bigl\{\langle\vect{a}, \vect{x}\rangle \equiv_2 0, \langle\vect{a}, \vect{x}\rangle \equiv_2 1\bigr\}$
\form#357:$\Gset_n$
\form#358:$\mu_1, \ldots, \mu_k \in \Zset$
\form#359:$\vect{v} = \sum_{j=1}^k \mu_j \vect{x}_j$
\form#360:$\inthull(S)$
\form#361:$\intaffinehull(S)$
\form#362:$\vect{p} \in \cL$
\form#363:$\vect{q} \in \Rset^n$
\form#364:$\vect{q} \neq \vect{0}$
\form#365:$\cL \neq \emptyset$
\form#366:$\vect{p} + \mu \vect{q} \in \cL$
\form#367:$\mu \in \Zset$
\form#368:$\vect{p} + \lambda \vect{l} \in \cL$
\form#369:$\lambda \in \Rset$
\form#370:$L, Q, P$
\form#371:\[ \cL = \linearhull(L) + \inthull(Q) + \intaffinehull(P) \]
\form#372:$\cL \in \Gset_n$
\form#373:$(L, Q, P)$
\form#374:$\cL = \ggen(L, Q, P)$
\form#375:$\cL = \ggen(L, Q, P) = \emptyset$
\form#376:$P \neq \emptyset$
\form#377:$\cL = \ggen(L, \emptyset, Q_{\vect{p}} \union P)$
\form#378:$\vect{p} \in P$
\form#379:$Q_{\vect{p}} = \{\, \vect{p} + \vect{q} \mid \vect{q} \in Q \,\}$
\form#380:$\cC'$
\form#381:$\card \cC \leq \card \cC'$
\form#382:$\cG = (L, Q, P)$
\form#383:$\cG' = (L', Q', P')$
\form#384:$\card L \leq \card L'$
\form#385:$\card Q + \card P \leq \card Q' + \card P'$
\form#386:$\pdim(\cG) = k$
\form#387:$0 \leq \pdim(\cG) \leq n$
\form#388:$x_k$
\form#389:$\mathrm{expr} = \langle \vect{a}, \vect{x} \rangle + b$
\form#390:$\bigl\{ (0, 0)^\transpose, (0, 3)^\transpose, (3, 0)^\transpose \bigr\}$
\form#391:$3x_0 + 2 x_1 + 1$
\form#392:$a_0 = 3, a_1 = 2, b = 1$
\form#393:$\cL_1$
\form#394:$\bigl\{ (1, 0)^\transpose, (7, 3)^\transpose, (10, 0)^\transpose \bigr\}$
\form#395:$(1, 0)$
\form#396:$(3, -3), (0, 9)$
\form#397:$\{x \equiv_3 1, x + y \equiv_9 1\}$
\form#398:$\cL_2$
\form#399:$x = y$
\form#400:$y$
\form#401:$\fund{\phi = (\mathrm{lhs}', \mathrm{rhs}, f)}{\Rset^n}{\Rset^n}$
\form#402:$f \in \Qset$
\form#403:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n \itc (\vect{v}, \vect{w}) \in \phi \iff \bigl( \langle \vect{c}, \vect{w} \rangle + d \equiv_f \langle \vect{a}, \vect{v} \rangle + b \bigr) \land \Bigl( \bigland_{0 \leq i < n, c_i = 0} w_i = v_i \Bigr). \]
\form#404:$\mathrm{expr} = \bigl(\langle \vect{a}, \vect{x} \rangle + b\bigr)$
\form#405:$c, f \in \Rset$
\form#406:$\cg = ( \mathrm{expr} \equiv_f c )$
\form#407:$\mathrm{expr}$
\form#408:$\mathrm{val} = \langle \vect{a}, \vect{w} \rangle + b$
\form#409:$\vect{w} \in \cL$
\form#410:\[ \lvert\mathrm{val}\rvert = \min\Bigl\{\, \bigl\lvert\langle \vect{a}, \vect{v} \rangle + b \bigr\rvert \Bigm| \vect{v} \in \cL \,\Bigr\}. \]
\form#411:$( \mathrm{expr} = c )$
\form#412:$0$
\form#413:$\cL_1, \cL_2 \in \Gset_n$
\form#414:$ \cL_1 \nearrow \cL_2$
\form#415:\[ \bigl\{\, \vect{p} + \mu \vect{q} \in \Rset^n \bigm| \vect{p} \in \cL_1, \vect{q} \in \cL_2, \mu \in \Zset \,\bigr\}. \]
\form#416:$ \cg = \bigl( \langle \vect{a}, \vect{x} \rangle \equiv_f b \bigr) $
\form#417:$\cL_{\cg} = \gcon\bigl(\{\cg\}\bigr)$
\form#418:$\cg$
\form#419:$\cL \inters \cL_{\cg} = \emptyset$
\form#420:$\cL \inters \cL_{\cg} \neq \emptyset$
\form#421:$\cL \inters \cL_{\cg} \subset \cL$
\form#422:$\cL \sseq \cL_{\cg}$
\form#423:$\cL \inters \cQ = \emptyset$
\form#424:$\cL \inters \cQ \neq \emptyset$
\form#425:$\cL \inters \cQ \subset \cL$
\form#426:$\cL \sseq \cQ$
\form#427:$=$
\form#428:$J$
\form#429:$\{0, \ldots, n-1\}$
\form#430:$w$
\form#431:$R = \{r \in \Rset \mid 0 \leq r < 2^w\}$
\form#432:$R = \{r \in \Rset \mid -2^{w-1} \leq r < 2^{w-1}\}$
\form#433:$v_j$
\form#434:$j$
\form#435:$\cL'$
\form#436:$a \in R$
\form#437:$a$
\form#438:$\cL' = \cL$
\form#439:$a + z \in R$
\form#440:$z \in \Zset$
\form#441:$(0, \ldots, 0, v_j, 0, \ldots, 0)$
\form#442:$v_j = 1$
\form#443:$v_j = a \mod 2^w$
\form#444:$a \in \Rset$
\form#445:$a'$
\form#446:$a' = a \mod 2^w$
\form#447:$a'\in R$
\form#448:$v_j = 2^w$
\form#449:$\cL_1 \widen \cL_2$
\form#450:$\cL_1 \sseq \cL_2$
\form#451:$D$
\form#452:$\mathord{\entails}$
\form#453:$\mathord{\meet}$
\form#454:$\true$
\form#455:$\false$
\form#456:$\cS \in \wp(D)$
\form#457:$\false \notin \cS$
\form#458:$\forall d_1, d_2 \in \cS \itc d_1 \entails d_2 \implies d_1 = d_2$
\form#459:$\wpfn{D}{\entails}$
\form#460:$\fund{\nonredmap}{\wpf(D)}{\wpfn{D}{\entails}}$
\form#461:$\cS \in \wpf(D)$
\form#462:\[ \nonredmap(\cS) \defeq \cS \setdiff \{\, d \in \cS \mid d = \false \text{ or } \exists d' \in \cS \st d \sentails d' \,\}. \]
\form#463:$d \sentails d'$
\form#464:$d \entails d' \land d \ne d'$
\form#465:$\cS$
\form#466:$\nonredmap(\cS)$
\form#467:$\cS \neq \{ \false \}$
\form#468:$D_{\smallP}$
\form#469:$\mathord{\entailsP}$
\form#470:$\cS_1$
\form#471:$\cS_2 \in D_{\smallP}$
\form#472:$\cS_1 \entailsP \cS_2$
\form#473:\[ \forall d_1 \in \cS_1 \itc \exists d_2 \in \cS_2 \st d_1 \entails d_2. \]
\form#474:$\{\true\}$
\form#475:$ \nonredmap \bigl( \{\, d_1 \meet d_2 \mid d_1 \in \cS_1, d_2 \in \cS_2 \,\} \bigr) $
\form#476:$\nonredmap(\cS_1 \union \cS_2)$
\form#477:$\cS \in D_{\smallP}$
\form#478:$d \in D$
\form#479:$\nonredmap\bigl(\cS \union \{d\}\bigr)$
\form#480:$\cS_1 = \{ d_1, \ldots, d_m \}$
\form#481:$\cS_2 = \{ c_1, \ldots, c_n \}$
\form#482:$\cS = \{ s_1, \ldots, s_q \}$
\form#483:$\cS_2$
\form#484:$q \leq m$
\form#485:$s_k \in \cS$
\form#486:$d_i \in \cS_1$
\form#487:$c_j \in \cS_2$
\form#488:$s_k$
\form#489:$d_i$
\form#490:$c_j$
\form#491:$\cS_1, \cS_2$
\form#492:$\uplus$
\form#493:$c, d$
\form#494:$c \uplus d \neq c \union d$
\form#495:$\cV$
\form#496:$\mathbb{F}_a$
\form#497:$\mathbb{F}_t$
\form#498:$-\infty$
\form#499:$+\infty$
\form#500:\[ i + \sum_{v \in \cV}i_{v}v, \]
\form#501:$v$
\form#502:$i_{v}$
\form#503:$\mathbb{I}_a$
\form#504:$\infty$
\form#505:$\mathbb{L}_a$
\form#506:\[ \left(i + \sum_{v \in V}i_{v}v\right) \aslf \left(i' + \sum_{v \in V}i'_{v}v\right) \defeq \left(i \asifp i'\right) + \sum_{v \in V}\left(i_{v} \asifp i'_{v}\right)v, \]
\form#507:\[ \left(i + \sum_{v \in V}i_{v}v\right) \adlf \left(i' + \sum_{v \in V}i'_{v}v\right) \defeq \left(i \adifp i'\right) + \sum_{v \in V}\left(i_{v} \adifp i'_{v}\right)v, \]
\form#508:\[ i \amlf \left(i' + \sum_{v \in V}i'_{v}v\right) \defeq \left(i \amifp i'\right) + \sum_{v \in V}\left(i \amifp i'_{v}\right)v, \]
\form#509:\[ \left(i + \sum_{v \in V}i_{v}v\right) \adivlf i' \defeq \left(i \adivifp i'\right) + \sum_{v \in V}\left(i_{v} \adivifp i'\right)v. \]
\form#510:$\asifp, \adifp, \amifp,$
\form#511:$\adivifp$
\form#512:$\fund{\rho^{\#}}{\cV}{\mathbb{I}_a}$
\form#513:$\fund{\rho^{\#}_l}{\cV}{\mathbb{L}_a}$
\form#514:$e$
\form#515:$\left \llbracket \rho^{\#}, \rho^{\#}_l \right \rrbracket$
\form#516:$\linexprenv{e}{\rho^{\#}}{\rho^{\#}_l}$
\form#517:$i+1$
\form#518:\[ \sum_{i=0}^{n-1} a_i x_i + b \]
\form#519:$x_i$
\form#520:$i + 1$
\form#521:$lf = x_5 - x_2 + 1$
\form#522:$4x - 2y - z + 14$
\form#523:$3$
\form#524:$\sum_{i=0}^{n-1} a_i x_i + b = 0$
\form#525:$\sum_{i=0}^{n-1} a_i x_i + b \geq 0$
\form#526:$\sum_{i=0}^{n-1} a_i x_i + b > 0$
\form#527:$3x + 5y - z = 0$
\form#528:$4x \geq 2y - 13$
\form#529:$2$
\form#530:$4x > 2y - 13$
\form#531:$x - 5y + 3z \leq 4$
\form#532:$x - 5y + 3z > 4$
\form#533:$\vect{l} = (a_0, \ldots, a_{n-1})^\transpose$
\form#534:$\vect{r} = (a_0, \ldots, a_{n-1})^\transpose$
\form#535:$\vect{p} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#536:$\vect{c} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#537:$d > 0$
\form#538:$x-y-z$
\form#539:$\vect{p} = (1, 0, 2)^\transpose \in \Rset^3$
\form#540:$\vect{0} \in \Rset^3$
\form#541:$\vect{0} \in \Rset^2$
\form#542:$\vect{0} \in \Rset^0$
\form#543:$\vect{p}$
\form#544:$\vect{q} = (-1.5, 3.2, 2.1)^\transpose \in \Rset^3$
\form#545:$\vect{c} = (1, 0, 2)^\transpose \in \Rset^3$
\form#546:$(a_0, \ldots, a_{n-1})^\transpose$
\form#547:$(a_0, 2 a_1, \ldots, (i+1)a_i, \ldots, n a_{n-1})^\transpose$
\form#548:$\vect{q} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#549:$\vect{p} = (1, -1, -1)^\transpose \in \Rset^3$
\form#550:$\vect{p1} = (-1.5, 3.2, 2.1)^\transpose \in \Rset^3$
\form#551:$\vect{q} = (1, 0, 2)^\transpose \in \Rset^3$
\form#552:$\cg = \sum_{i=0}^{n-1} a_i x_i + b = 0 \pmod m$
\form#553:$m \neq 0$
\form#554:$\pmod{1}$
\form#555:$4x = 2y - 13 \pmod{1}$
\form#556:$4x = 2y - 13 \pmod{2}$
\form#557:$x - 5y + 3z = 4 \pmod{5}$
\form#558:$2 e$
\form#559:$2x - 10y + 6z = 8 \pmod{5}$
\form#560:$1$
\form#561:$x + y = 1 \pmod{2}$
\form#562:$\Zset^2$
\form#563:$(0, 1)^\transpose \in \Rset^2$
\form#564:\[ \bigl\{\, (x, 0)^\transpose \in \Rset^2 \bigm| x \geq 0 \,\bigr\}, \]
\form#565:$R^0$
\form#566:$\{ 2 \} \sseq \Rset$
\form#567:\[ \bigl\{\, (2, y)^\transpose \in \Rset^2 \bigm| y \in \Rset \,\bigr\}. \]
\form#568:$\bigl\{ (2, 0)^\transpose \bigr\} \sseq \Rset^2$
\form#569:$x+4$
\form#570:$x+y$
\form#571:$x-y$
\form#572:$\bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2$
\form#573:$\bigl\{(3, 0)^\transpose \bigr\} \sseq \Rset^2$
\form#574:\[ \bigl\{\, (x, 0)^\transpose \in \Rset^2 \bigm| x \pmod{1}\ 0 \,\bigr\}, \]
\form#575:$x + y = 2$
\form#576:$x+3$
\form#577:$(x, y)$
\form#578:$x=y$
\form#579:$x=-y$
\form#580:$\pm x_i \leq k$
\form#581:$x_i - x_j \leq k$
\form#582:\[ a_i x_i - a_j x_j \relsym b \]
\form#583:$\mathord{\relsym} \in \{ \leq, =, \geq \}$
\form#584:$a_j$
\form#585:$a_i = 0$
\form#586:$a_j = 0$
\form#587:$a_i = a_j$
\form#588:$3x - 3y \leq 1$
\form#589:$x - y \leq 1$
\form#590:$x - y \leq \frac{1}{3}$
\form#591:$x - y \leq k$
\form#592:$k > \frac{1}{3}$
\form#593:$\frac{1}{3}$
\form#594:$3x - y \leq 1$
\form#595:$\Rset^3$
\form#596:\[ ax_i + bx_j \leq k \]
\form#597:$a, b \in \{-1, 0, 1\}$
\form#598:\[ \pm a_i x_i \pm a_j x_j \relsym b \]
\form#599:$3x + 3y \leq 1$
\form#600:$x + y \leq 1$
\form#601:$x + y \leq \frac{1}{3}$
\form#602:$x + y \leq k$
\form#603:$\perp$
\form#604:$M$
\form#605:$M-x'_i$
\form#606:$x'$
\form#607:$x'_i = M-x_i$
\form#608:$x_i = M-x'_i$
\form#609:$(x,y)$
\form#610:\[\left\{\begin{array}{l} y \geq 2x - 4\\ y \leq -x + p \end{array}\right.\]
\form#611:\[\left\{\begin{array}{l} M - y \geq 2M - 2x - 4\\ M - y \leq -M + x + p \end{array}\right.\]
\form#612:\[ \left\{\begin{array}{l} x' = M - \left\lfloor\frac{p+1}{3}\right\rfloor - 1 \\ y' = M - p + \left\lfloor\frac{p+1}{3}\right\rfloor + 1 \end{array}\right. \]
\form#613:\[ \left\{\begin{array}{l} x = \left\lfloor\frac{p+1}{3}\right\rfloor + 1 \\ y = p - \left\lfloor\frac{p+1}{3}\right\rfloor - 1 \end{array}\right. \]
\form#614:$x'_i-M$
\form#615:$x'_i = x_i+M$
\form#616:$x_i = x'_i-M$
\form#617:\[\left\{\begin{array}{l} y \geq -2x - 4\\ 2y \leq x + 2p \end{array}\right.\]
\form#618:\[\left\{\begin{array}{l} y' - M \geq -2x' + 2M - 4\\ 2y' - 2M \leq x' - M + 2p \end{array}\right.\]
\form#619:\[ \left\{\begin{array}{l} x' = M - \left\lfloor\frac{2p+3}{5}\right\rfloor - 1 \\ y' = M + 2\left\lfloor\frac{2p+3}{5}\right\rfloor - 2 \end{array}\right. \]
\form#620:\[ \left\{\begin{array}{l} x = -\left\lfloor\frac{2p+3}{5}\right\rfloor - 1 \\ y = 2\left\lfloor\frac{2p+3}{5}\right\rfloor - 2 \end{array}\right. \]
\form#621:$p^+-p^-$
\form#622:$p^+$
\form#623:$p^-$
\form#624:$p_i$
\form#625:$p^+_i-p^-$
\form#626:$-p^-$
\form#627:$x_1, x_2, \dots, x_n$
\form#628:$p_1, p_2, \dots, p_m$
\form#629:$f(x_2, \dots, x_n, p_1, \dots, p_m)$
\form#630:$x_1 \geq f(x_2, \dots, x_n, p_1, \dots, p_m)$
\form#631:$D_1$
\form#632:$D_2$
\form#633:$\fund{\gamma_1}{D_1}{\Rset^n}$
\form#634:$\fund{\gamma_2}{D_2}{\Rset^n}$
\form#635:$D = D_1 \times D_2$
\form#636:$\fund{\gamma}{D}{\Rset^n}$
\form#637:$d = (d_1, d_2) \in D$
\form#638:\[ \gamma(d) = \gamma_1(d_1) \inters \gamma_2(d_2). \]
\form#639:$d = (G, P) \in (\Gset \times \Pset)$
\form#640:$G$
\form#641:$[-\omega, \omega]$
\form#642:$\omega$
\form#643:$u \cdot 2^s$
\form#644:$u$
\form#645:$s$
\form#646:$\varepsilon_{\mathbf{f}}(l)$
\form#647:\[ \varepsilon_{\mathbf{f}}\left([a, b]+\sum_{v \in \cV}[a_{v}, b_{v}]v\right) \defeq (\textrm{max}(|a|, |b|) \amifp [-\beta^{-\textrm{p}}, \beta^{-\textrm{p}}]) + \sum_{v \in \cV}(\textrm{max}(|a_{v}|,|b_{v}|) \amifp [-\beta^{-\textrm{p}}, \beta^{-\textrm{p}}])v \]
\form#648:$\mathbf{f}$
\form#649:$\iota(lf)\rho^{\#}$
\form#650:\[ \iota\left(i + \sum_{v \in \cV}i_{v}v\right)\rho^{\#} \defeq i \asifp \left(\bigoplus_{v \in \cV}{}^{\#}i_{v} \amifp \rho^{\#}(v)\right) \]
\form#651:$\rho^{\#}(v)$
\form#652:$ y \cdot 2^\mathtt{exp} $
\form#653:$ y / 2^\mathtt{exp} $
\form#654:$15 \cdot 10^2 = 1500$
\form#655:$5$
\form#656:$3/580000$
\form#657:$713/10$
\form#658:$-1929/15625$
\form#659:$11/50$
\form#660:$-220001/100$
\form#661:$35$
\form#662:$44027$
\form#663:$7/2$
\form#664:$2000$
\form#665:$1024$
\form#666:$1088$
\form#667:$9073863231288$
\form#668:$17/8$
\form#669:$482$
\form#670:$256$
\form#671:\[ \{\, a \in \mathtt{to} \mid \exists b \in \mathtt{x} \st a \mathrel{\mathtt{rel}} b \,\}. \]
\form#672:\[ \{\, a \in \mathtt{to} \mid \forall b \in \mathtt{x} \itc a \mathrel{\mathtt{rel}} b \,\}. \]
\form#673:$O(n)$
\form#674:$O(1)$
\form#675:$O(n*\log^2 n)$
\form#676:$O(\log n)$
\form#677:$O(\log^2 n)$
\form#678:$O(k+\log^2 n)$
\form#679:$O(k+\log n)$
\form#680:$O(\log(last - first + 1))$
\form#681:$O(k)$
\form#682:$O(k*\log^2 n)$
\form#683:$O(k + \log m)$
\form#684:$ \{ x_1 \mapsto x_2, x_2 \mapsto x_3, x_3 \mapsto x_1 \}$
\form#685:$ x_1, x_2, x_3 $
\form#686:$cg = \bigl(\sum_{i=0}^{n-1} a_i x_i + b = 0 \pmod{m}\bigr)$
\form#687:$\sum_{i=0}^{space_dim-1} a_i x_i + b$
\form#688:$c = \bigl(\sum_{i=0}^{n-1} a_i x_i + b \relsym 0\bigr)$
\form#689:$\sum_{i=0}^{n-1} a_i x_i + b$
\form#690:$g = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#691:$d = 1$
\form#692:$\sum_{i=0}^{n-1} a_i x_i$
\form#693:$0<=0$
\form#694:$0 \leq 1$
\form#695:$\sum_{i=0}^{n-1} 0 x_i + 0 = 0$
\form#696:$\sum_{i=0}^{n-1} 0 x_i + b \geq 0$
\form#697:$b \geq 0$
\form#698:$\sum_{i=0}^{n-1} 0 x_i + b > 0$
\form#699:$b > 0$
\form#700:$\sum_{i=0}^{n-1} 0 x_i + b = 0$
\form#701:$b \neq 0$
\form#702:$b < 0$
\form#703:$b \leq 0$
\form#704:$\epsilon \geq 0$
\form#705:$\epsilon \leq 1$
\form#706:$0<0$
\form#707:$L_\infty$
\form#708:$ e_1 = e_2 \pmod{m}$
\form#709:$ e_1 = e_2 \pmod{mk}$
\form#710:$\sum_{i=0}^{n-1} 0 x_i + 0 == 0$
\form#711:$\sum_{i=0}^{n-1} 0 x_i + b \%= 0 / m$
\form#712:$b = 0 \pmod{m}$
\form#713:$\sum_{i=0}^{n-1} 0 x_i + b == 0$
\form#714:$b \neq 0 \pmod{m}$
\form#715:$0 = 1 \pmod{1}$
\form#716:$0 = 1 \pmod{0}$
\form#717:$e1 = e2 \pmod{1}$
\form#718:$e = n \pmod{1}$
\form#719:$n = e \pmod{1}$
\form#720:$\mathrm{var}' \relsym \frac{\mathrm{expr}}{\mathrm{denominator}}$
\form#721:$\mathrm{lhs}' \relsym \mathrm{rhs}$
\form#722:$\frac{\mathrm{lb\_expr}}{\mathrm{denominator}} \leq \mathrm{var}' \leq \frac{\mathrm{ub\_expr}}{\mathrm{denominator}}$
\form#723:$\mathtt{y} \Delta \mathtt{x}$
\form#724:$\cB \sseq \Rset^2$
\form#725:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cB \,\bigr\}. \]
\form#726:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cB \,\bigr\}. \]
\form#727:$B \sseq \Rset^n$
\form#728:$D \sseq \Rset^m$
\form#729:$R \sseq \Rset^{n+m}$
\form#730:\[ R \defeq \Bigl\{\, (x_1, \ldots, x_n, y_1, \ldots, y_m)^\transpose \Bigm| (x_1, \ldots, x_n)^\transpose \in B, (y_1, \ldots, y_m)^\transpose \in D \,\Bigl\}. \]
\form#731:$f(k)$
\form#732:$\dots$
\form#733:$I$
\form#734:$d$
\form#735:$n/d$
\form#736:$\times$
\form#737:$\epsilon$
\form#738:$M \in \Rset^r \times \Rset^c$
\form#739:$N \in \Rset^{r+n} \times \Rset^{c+n}$
\form#740:$N = \bigl(\genfrac{}{}{0pt}{}{0}{M}\genfrac{}{}{0pt}{}{J}{o}\bigr)$
\form#741:$n \times n$
\form#742:$\sum_{i = 0}^{n - 1} a_i x_i + b$
\form#743:${a'}_{ij}$
\form#744:$a_{ij}$
\form#745:\[ {a'}_{ij} = \begin{cases} a_{ij} * \mathrm{denominator} + a_{iv} * \mathrm{expr}[j] \quad \text{for } j \neq v; \\ \mathrm{expr}[v] * a_{iv} \quad \text{for } j = v. \end{cases} \]
\form#746:$ e_1 = e_2 $
\form#747:$ e_1 = e_2 \pmod{0}$
\form#748:$r \times c$
\form#749:$(r+dims) \times (c+dims)$
\form#750:$\bigl(\genfrac{}{}{0pt}{}{0}{A} \genfrac{}{}{0pt}{}{B}{A}\bigr)$
\form#751:$B$
\form#752:$dims \times dims$
\form#753:$\bigl(\genfrac{}{}{0pt}{}{0}{1} \genfrac{}{}{0pt}{}{1}{0}\bigr)$
\form#754:\[ \frac{\sum_{i = 0}^{n - 1} a_i x_i + b} {\mathrm{denominator}}. \]
\form#755:$\leq$
\form#756:$\relsym$
\form#757:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cP \,\bigr\}. \]
\form#758:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cP \,\bigr\}. \]
\form#759:$\Rset^0 = \{\cdot\}$
\form#760:\[ \begin{cases} sat\_c[i][j] = 0, \quad \text{if } G[i] \cdot C^\mathrm{T}[j] = 0; \\ sat\_c[i][j] = 1, \quad \text{if } G[i] \cdot C^\mathrm{T}[j] > 0. \end{cases} \]
\form#761:\[ \begin{cases} sat\_g[i][j] = 0, \quad \text{if } C[i] \cdot G^\mathrm{T}[j] = 0; \\ sat\_g[i][j] = 1, \quad \text{if } C[i] \cdot G^\mathrm{T}[j] > 0. \end{cases} \]
\form#762:$O(r)$
\form#763:$O(n+k)$
\form#764:$O(\sum_{j=1}^{r} (k_j*\log^2 n_j))$
\form#765:$k_j$
\form#766:$n_j$
\form#767:$O(r+k*\log^2 c)$
\form#768:$(r+n) \times (c+m)$
\form#769:$\bigl(\genfrac{}{}{0pt}{}{M}{0} \genfrac{}{}{0pt}{}{0}{0}\bigr)$
\form#770:$(r+n) \times c$
\form#771:$\genfrac{(}{)}{0pt}{}{M}{0}$
\form#772:$(r+1) \times c$
\form#773:$\genfrac{(}{)}{0pt}{}{M}{y}$
\form#774:$ \{ 1 \mapsto 3, 2 \mapsto 4, 3 \mapsto 6, 4 \mapsto 2, 5 \mapsto 5, 6 \mapsto 1 \}$
\form#775:$(1 3 6)(2 4)$
\form#776:$O(k*\sum_{j=1}^{r} \log^2 n_j)$
\form#777:$O(k*r*\log^2 c)$
\form#778:$r \times (c+n)$
\form#779:$(M \, 0)$
\form#780:$O(\sum_{j=1}^{r} (k_j+\log n_j))$
\form#781:$O(k+r*\log c)$
\form#782:$O(k + \sum_{j=1}^{r} (\log^2 n_j))$
\form#783:$O(r*(c-i+\log^2 c))$
\form#784:$O(\sum_{j=1}^r (k_j*\log n_j))$
\form#785:$O(r*n*\log c)$
\form#786:$O(n*\log n)$
\form#787:$O(r+k)$
\form#788:$[\mathrm{first}, \mathrm{last})$
\form#789:$\frac{\mathtt{numer}}{\mathtt{denom}}$
\form#790:$\frac{d_{j}}{\|\Delta x^{j} \|}$
\form#791:\[ \|\Delta x^{j} \| = \left( 1+\sum_{i=1}^{m} \alpha_{ij}^2 \right)^{\frac{1}{2}}. \]
\form#792:$\alpha$
\form#793:$\rho^{\#}$
\form#794:$\rho^{\#}_l$
\form#795:$(C, G)$
\form#796:\[ C\vect{x} \geq \vect{0} \quad\iff\quad \exists \vect{\lambda} \geq \vect{0} \mathrel{.} \vect{x} = G\vect{\lambda}. \]
\form#797:\[ P = \{\, \vect{x} \in \Rset^n \mid C\vect{x} \geq \vect{0}\, \} = \{\, \vect{x} \in \Rset^n \mid \vect{x} = G\vect{\lambda} \text{ for some } \vect{\lambda} \geq \vect{0}\, \}. \]
\form#798:$m \times n$
\form#799:$\mathop{\mathrm{rank}}(C) = n = \mathit{dimension of the whole space}$
\form#800:$A \in \Rset^{m \times n}$
\form#801:$\{ \vect{a}_1, \ldots, \vect{a}_m \}$
\form#802:$\vect{y} \in \Rset^n$
\form#803:$\vect{a}_i$
\form#804:$\vect{c}$
\form#805:$ \forall \vect{y} \mathrel{:} (A\vect{y} \geq 0 \implies \langle \vect{y},\vect{c} \rangle \geq 0) $
\form#806:$ \exists \vect{\lambda} \geq \vect{0} \mathrel{.} \vect{c}^\mathrm{T} = \vect{\lambda}^\mathrm{T}A $
\form#807:$(G^\mathrm{T}, C^\mathrm{T})$
\form#808:$C\vect{x} \geq \vect{0}$
\form#809:$\exists \lambda \geq 0 \mathrel{.} \vect{x} = G\vect{\lambda}$
\form#810:$ \exists \vect{\lambda} \geq \vect{0} \mathrel{.} \vect{x}^\mathrm{T} = \vect{\lambda}^\mathrm{T}G^\mathrm{T} $
\form#811:$\vect{z}$
\form#812:$G^\mathrm{T}\vect{z} \geq \vect{0}$
\form#813:$\exists \vect{\mu} \geq \vect{0} \mathrel{.} \vect{z} = C^\mathrm{T}\vect{\mu}$
\form#814:$\vect{x}$
\form#815:$ \forall \vect{y} \mathrel{:} (G^\mathrm{T}\vect{y} \geq \vect{0} \implies \langle \vect{y}, \vect{x} \rangle \geq 0) $
\form#816:$\vect{y} = \vect{z}$
\form#817:$\langle \vect{z}, \vect{x} \rangle \geq 0$
\form#818:$\langle \vect{x}, \vect{z} \rangle \geq 0$
\form#819:$ \forall \vect{x} \mathrel{:} (C\vect{x} \geq \vect{0} \implies \langle \vect{x}, \vect{z} \rangle \geq 0) $
\form#820:$\exists \vect{\mu} \geq \vect{0} \mathrel{.} \vect{z}^\mathrm{T} = \vect{\mu}^\mathrm{T} C$
\form#821:$CG \geq 0$
\form#822:$\vect{e}_j$
\form#823:$\vect{e}_j \geq \vect{0}$
\form#824:$\vect{\lambda} = \vect{e}_j$
\form#825:$\vect{x} = G\vect{\lambda} = G \vect{e}_j$
\form#826:$C\vect{x} = C(G \vect{e}_j) = (CG)\vect{e}_j \geq \vect{0}$
\form#827:$(CG)\vect{e}_j$
\form#828:$CG$
\form#829:$CG \geq \vect{0}$
\form#830:$\exists \vect{\mu} \geq \vect{0} \mathrel{.} \vect{z}^\mathrm{T} = \vect{\mu}^\mathrm{T}C$
\form#831:$\forall \vect{y} \mathrel{:} (C\vect{y} \geq \vect{0} \implies \langle \vect{y}, \vect{z} \rangle \geq 0)$
\form#832:$\vect{y} = G\vect{e}_j$
\form#833:$C\vect{y} = CG\vect{e}_j \geq 0$
\form#834:$ \langle \vect{y}, \vect{z} \rangle = (\vect{e}_j^\mathrm{T}G^\mathrm{T}) \vect{z} = \vect{e}_j^\mathrm{T}(G^\mathrm{T} \vect{z}) \geq 0 $
\form#835:$G^\mathrm{T}\vect{z}$
\form#836:$\vect{y}_1$
\form#837:$\vect{y}_2$
\form#838:\[ \vect{y}_2 = \lambda \vect{y}_1 + \mu \vect{z} \]
\form#839:$\vect{c}_1$
\form#840:\[ \langle \vect{c}_1, \vect{y}_2 \rangle = \langle \vect{c}_1, (\lambda \vect{y}_1 + \mu \vect{z}) \rangle = \lambda \langle \vect{c}_1, \vect{y}_1 \rangle + \mu \langle \vect{c}_1, \vect{z} \rangle = 0 + \mu \langle \vect{c}_1, \vect{z} \rangle = \mu \langle \vect{c}_1, \vect{z} \rangle \]
\form#841:\[ \mu \langle \vect{c}_1, \vect{z} \rangle = 0. \]
\form#842:$d - 2$
\form#843:$F$
\form#844:\[ \vect{x} = \lambda \vect{r}_1 + \mu \vect{r}_2. \]
\form#845:\[ \langle \vect{c}, \vect{x} \rangle \geq 0 \]
\form#846:\[ \langle \vect{c}, \vect{x} \rangle = \lambda \langle \vect{c}, \vect{r}_1 \rangle (or = \mu \langle \vect{c}, \vect{r}_2 \rangle). \]
\form#847:$d - k$
\form#848:$\overline Q$
\form#849:$d - 1$
\form#850:\[ \langle \vect{c}, \vect{r}_1 \rangle = 0 \land \langle \vect{c}, \vect{r}_2 \rangle = 0 \Rightarrow \langle \vect{c}, (\lambda_1 \vect{r}_1 + \lambda_2 \vect{r}_2) \rangle = \lambda_1 \langle \vect{c}, \vect{r}_1 \rangle + \lambda_2 \langle \vect{c}, \vect{r}_2 \rangle = 0, \]
\form#851:$\lambda_1, \lambda_2$
\form#852:$\bigl(\genfrac{}{}{0pt}{}{A}{0} \genfrac{}{}{0pt}{}{0}{B}\bigr)$
\form#853:$\bigl(\genfrac{}{}{0pt}{}{1}{0} \genfrac{}{}{0pt}{}{0}{1}\bigr)$
\form#854:$\mathrm{var}' = \frac{\mathrm{expr}}{\mathrm{denominator}} \pmod{\mathrm{modulus}}$
\form#855:$\mathrm{lhs}' = \mathrm{rhs} \pmod{\mathrm{modulus}}$
\form#856:$\cL \sseq \Rset^2$
\form#857:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cL \,\bigr\}. \]
\form#858:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cL \,\bigr\}. \]
\form#859:$\{3x \equiv_3 0, 4x + y \equiv_3 1\}$
\form#860:$\{3x \equiv_3 0, x + y \equiv_3 1\}$
\form#861:$\mathrm{var} \relsym \frac{\mathrm{expr}}{\mathrm{denominator}}$
\form#862:$\cO \sseq \Rset^2$
\form#863:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cO \,\bigr\}. \]
\form#864:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cO \,\bigr\}. \]
\form#865:$O(n^2)$
\form#866:$i + \sum_{v \in \cV}i_{v}v $
\form#867:$i' + \sum_{v \in \cV}i'_{v}v $
\form#868:$\aslf$
\form#869:\[ \left(i + \sum_{v \in \cV}i_{v}v \right) \aslf \left(i' + \sum_{v \in \cV}i'_{v}v \right) = \left(i \asifp i'\right) + \sum_{v \in \cV}\left(i_{v} \asifp i'_{v} \right)v. \]
\form#870:$e_{1} \oplus e_{2}$
\form#871:$\linexprenv{e_{1} \oplus e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\form#872:\[ \linexprenv{e_{1} \oplus e_{2}}{\rho^{\#}}{\rho^{\#}_l} = \linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \aslf \linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l} \aslf \varepsilon_{\mathbf{f}}\left(\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \right) \aslf \varepsilon_{\mathbf{f}}\left(\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l} \right) \aslf mf_{\mathbf{f}}[-1, 1] \]
\form#873:$l$
\form#874:$mf_{\mathbf{f}}$
\form#875:$\adlf$
\form#876:\[ \left(i + \sum_{v \in \cV}i_{v}v\right) \aslf \left(i' + \sum_{v \in \cV}i'_{v}v\right) = \left(i \asifp i'\right) + \sum_{v \in \cV}\left(i_{v} \asifp i'_{v}\right)v, \]
\form#877:\[ \left(i + \sum_{v \in \cV}i_{v}v\right) \adlf \left(i' + \sum_{v \in \cV}i'_{v}v\right) = \left(i \adifp i'\right) + \sum_{v \in \cV}\left(i_{v} \adifp i'_{v}\right)v. \]
\form#878:$e_{1} \ominus e_{2}$
\form#879:$\linexprenv{e_{1} \ominus e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\form#880:\[ \linexprenv{e_{1} \ominus e_{2}}{\rho^{\#}}{\rho^{\#}_l} = \linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \adlf \linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l} \aslf \varepsilon_{\mathbf{f}}\left(\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \right) \aslf \varepsilon_{\mathbf{f}}\left(\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l} \right) \aslf mf_{\mathbf{f}}[-1, 1] \]
\form#881:$\amlf$
\form#882:\[ i \amlf \left(i' + \sum_{v \in \cV}i'_{v}v\right) = \left(i \amifp i'\right) + \sum_{v \in \cV}\left(i \amifp i'_{v}\right)v. \]
\form#883:$[a, b] \otimes e_{2}$
\form#884:$\linexprenv{[a, b] \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\form#885:\[ \linexprenv{[a, b] \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l} = \left([a, b] \amlf \linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l}\right) \aslf \left([a, b] \amlf \varepsilon_{\mathbf{f}}\left(\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l} \right)\right) \aslf mf_{\mathbf{f}}[-1, 1]. \]
\form#886:$e_{1} \otimes [a, b]$
\form#887:$\linexprenv{e_{1} \otimes [a, b]}{\rho^{\#}}{\rho^{\#}_l}$
\form#888:\[ \linexprenv{e_{1} \otimes [a, b]}{\rho^{\#}}{\rho^{\#}_l} = \linexprenv{[a, b] \otimes e_{1}}{\rho^{\#}}{\rho^{\#}_l}. \]
\form#889:$e_{1} \otimes e_{2}$
\form#890:$\linexprenv{e_{1} \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\form#891:\[ \linexprenv{e_{1} \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l} = \linexprenv{\iota\left(\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \right)\rho^{\#} \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l}, \]
\form#892:$\iota(l)\rho^{\#}$
\form#893:$\adivlf$
\form#894:\[ \left(i + \sum_{v \in \cV}i_{v}v\right) \adivlf i' = \left(i \adivifp i'\right) + \sum_{v \in \cV}\left(i_{v} \adivifp i'\right)v. \]
\form#895:$e_{1} \oslash [a, b]$
\form#896:$ \linexprenv{e_{1} \oslash [a, b]}{\rho^{\#}}{\rho^{\#}_l} $
\form#897:\[ \linexprenv{e_{1} \oslash [a, b]}{\rho^{\#}}{\rho^{\#}_l} = \left(\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \adivlf [a, b]\right) \aslf \left(\varepsilon_{\mathbf{f}}\left( \linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \right) \adivlf [a, b]\right) \aslf mf_{\mathbf{f}}[-1, 1], \]
\form#898:$e_{1} \oslash e_{2}$
\form#899:$\linexprenv{e_{1} \oslash e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\form#900:\[ \linexprenv{e_{1} \oslash e_{2}}{\rho^{\#}}{\rho^{\#}_l} = \linexprenv{e_{1} \oslash \iota\left( \linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l} \right)\rho^{\#}}{\rho^{\#}}{\rho^{\#}_l}, \]
\form#901:$\frac{\mathtt{expr}}{\mathtt{d}}$
\form#902:$f_i(x,p) = 0 ; 1 \leq i \leq n$
\form#903:$f_i(x,p) \geq 0 ; 1 \leq i \leq n$
\form#904:$\sum\limits_{i=1}^n f_i(x,p) \leq 0$
\form#905:$c=s_{*j}\frac{t_{ik}}{s_{ij}}$
\form#906:$c'=s_{*j'}\frac{t_{i'k}}{s_{i'j'}}$
\form#907:$s_{*j}$
\form#908:$s_{*j'}$
\form#909:$j'$
\form#910:$(i,j)$
\form#911:$c'$
\form#912:$(i',j')$
\form#913:$-c$
\form#914:$-c'$
\form#915:\[ \mathit{ps} = \bigl\{ \{ x \geq 0 \}, \{ x \leq 0 \} \bigr\}, \]
\form#916:$\mathit{ps}$
\form#917:$\cP \uplus \cQ \neq \cP \union \cQ$
\form#918:$ x'_1, \ldots, x'_n $
\form#919:$ 0, \ldots, n-1 $
\form#920:$ x_1, \ldots, x_n $
\form#921:$ n, \ldots, 2n-1 $
\form#922:$ n+1 $
\form#923:$ \mu_0 + \sum_{i=1}^n \mu_i x_i $
\form#924:$ \mu_0, \mu_1, \ldots, \mu_n $
\form#925:$ n, 0, \ldots, n-1 $
\form#926:$\linexprenv{cast(e)}{\rho^{\#}}{\rho^{\#}_l}$
\form#927:\[ \linexprenv{cast(e)}{\rho^{\#}}{\rho^{\#}_l} = \linexprenv{e}{\rho^{\#}}{\rho^{\#}_l} \aslf \varepsilon_{\mathbf{f}}\left(\linexprenv{e}{\rho^{\#}}{\rho^{\#}_l} \right) \aslf mf_{\mathbf{f}}[-1, 1] \]
\form#928:$[l, u]$
\form#929:$\linexprenv{v}{\rho^{\#}}{\rho^{\#}_l}$
\form#930:$\rho^{\#}_l(v)$
\form#931:$[-1, 1]v$
\form#932:$\oplus$
\form#933:$\ominus$
\form#934:$\otimes$
\form#935:$\oslash$
\form#936:\[ \adlf \left(i + \sum_{v \in \cV}i_{v}v\right) = \left(\adifp i\right) + \sum_{v \in \cV}\left(\adifp i_{v} \right)v, \]
\form#937:$\ominus e$
\form#938:$\linexprenv{\ominus e}{\rho^{\#}}{\rho^{\#}_l}$
\form#939:\[ \linexprenv{\ominus e}{\rho^{\#}}{\rho^{\#}_l} = \adlf \left( \linexprenv{e}{\rho^{\#}}{\rho^{\#}_l} \right). \]
