\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 \nearrow_{>0} \cQ$
\form#287:\[ \bigl\{\, \vect{p} + \lambda \vect{q} \in \Rset^n \bigm| \vect{p} \in \cP, \vect{q} \in \cQ, \lambda \in \Rset^{>0} \,\bigr\}, \]
\form#288:$\Rset^{>0}$
\form#289:$\cP \nearrow \cQ$
\form#290:$\cP \nearrow \cQ = \cP \cup \left(\cP \nearrow_{>0} \cQ\right)$
\form#291:$\cP, \cQ, \cR \in \Pset_n$
\form#292:$\cR$
\form#293:$\cR \inters \cQ = \cP \inters \cQ$
\form#294:$\cR \Sseq \cP$
\form#295:$r \leq p$
\form#296:$r$
\form#297:$p$
\form#298:$ c = \bigl( \langle \vect{a}, \vect{x} \rangle \relsym b \bigr) $
\form#299:$c$
\form#300:$\cP \inters \cQ = \emptyset$
\form#301:$\cP \inters \cQ \neq \emptyset$
\form#302:$\cP \inters \cQ \subset \cP$
\form#303:$\cP \sseq \cQ$
\form#304:$\cP \sseq \cH$
\form#305:$\cH$
\form#306:$g$
\form#307:$\cP \widen \cQ$
\form#308:$\cP, \cQ \in \CPset_n$
\form#309:$\cP \Sseq \cP \widen \cQ$
\form#310:$\cQ \sseq \cP$
\form#311:$\cP = \cQ$
\form#312:$k$
\form#313:$\mathord{\relsym} \in \{ =, \geq, >\}$
\form#314:$k \in \{ 0, \ldots, i-1, i+1, \ldots, n-1 \}$
\form#315:$a_k = 0$
\form#316:$x = r$
\form#317:$x \leq r$
\form#318:$x \geq r$
\form#319:$x < r$
\form#320:$x > r$
\form#321:$r \in \Rset$
\form#322:$\cB$
\form#323:$\vect{e}_i = (0, \ldots, 1, \ldots, 0)^\transpose$
\form#324:$\langle \vect{e}_i, \vect{x} \rangle \relsym b$
\form#325:$\mathord{\relsym}$
\form#326:$\mathord{\geq}$
\form#327:$\mathord{>}$
\form#328:$\langle\vect{e}_i,\vect{x}\rangle \relsym b$
\form#329:$\mathord{\leq}$
\form#330:$\mathord{<}$
\form#331:$\{-2, -1, 0, 1, 2\}$
\form#332:$\mathord{\relsym} \in \{ =, \geq\}$
\form#333:$i, j \in \{ 0, \ldots, n-1 \}$
\form#334:$a_i, a_j \in \{ -1, 0, 1 \}$
\form#335:$a_i \neq a_j$
\form#336:$k \notin \{ i, j \}$
\form#337:$a, b, f \in \Rset$
\form#338:$a \equiv_f b$
\form#339:$\exists \mu \in \Zset \st a - b = \mu f$
\form#340:$\Sset \in \{ \Qset, \Rset \}$
\form#341:$\vect{a} \in \Sset^n \setdiff \{\vect{0}\}$
\form#342:$b, f \in \Sset$
\form#343:$\langle \vect{a}, \vect{x} \rangle \equiv_f b$
\form#344:$\Sset^n$
\form#345:\[ \bigl\{\, \vect{v} \in \Rset^n \bigm| \exists \mu \in \Zset \st \langle \vect{a}, \vect{v} \rangle = b + \mu f \,\bigr\}; \]
\form#346:$f \neq 0$
\form#347:$\langle \vect{a}, \vect{x} \rangle \equiv_0 b$
\form#348:$f = 0$
\form#349:$f$
\form#350:\[ \big\{\, \bigl(\langle \vect{a}, \vect{x} \rangle = b + \mu f\bigr) \bigm| \mu \in \Zset \,\bigr\}; \]
\form#351:$b \equiv_f 0$
\form#352:$\langle \vect{0}, \vect{x} \rangle \equiv_f b$
\form#353:$\cL \sseq \Rset^n$
\form#354:$\cL$
\form#355:$\Qset^n$
\form#356:$\cL = \emptyset$
\form#357:$\Gset_{n}$
\form#358:$\bigl\{\langle\vect{0}, \vect{x}\rangle \equiv_0 1\bigr\}$
\form#359:$0 = 1$
\form#360:$\bigl\{\langle\vect{a}, \vect{x}\rangle \equiv_2 0, \langle\vect{a}, \vect{x}\rangle \equiv_2 1\bigr\}$
\form#361:$\Gset_n$
\form#362:$\mu_1, \ldots, \mu_k \in \Zset$
\form#363:$\vect{v} = \sum_{j=1}^k \mu_j \vect{x}_j$
\form#364:$\inthull(S)$
\form#365:$\intaffinehull(S)$
\form#366:$\vect{p} \in \cL$
\form#367:$\vect{q} \in \Rset^n$
\form#368:$\vect{q} \neq \vect{0}$
\form#369:$\cL \neq \emptyset$
\form#370:$\vect{p} + \mu \vect{q} \in \cL$
\form#371:$\mu \in \Zset$
\form#372:$\vect{p} + \lambda \vect{l} \in \cL$
\form#373:$\lambda \in \Rset$
\form#374:$L, Q, P$
\form#375:\[ \cL = \linearhull(L) + \inthull(Q) + \intaffinehull(P) \]
\form#376:$\cL \in \Gset_n$
\form#377:$(L, Q, P)$
\form#378:$\cL = \ggen(L, Q, P)$
\form#379:$\cL = \ggen(L, Q, P) = \emptyset$
\form#380:$P \neq \emptyset$
\form#381:$\cL = \ggen(L, \emptyset, Q_{\vect{p}} \union P)$
\form#382:$\vect{p} \in P$
\form#383:$Q_{\vect{p}} = \{\, \vect{p} + \vect{q} \mid \vect{q} \in Q \,\}$
\form#384:$\cC'$
\form#385:$\card \cC \leq \card \cC'$
\form#386:$\cG = (L, Q, P)$
\form#387:$\cG' = (L', Q', P')$
\form#388:$\card L \leq \card L'$
\form#389:$\card Q + \card P \leq \card Q' + \card P'$
\form#390:$\pdim(\cG) = k$
\form#391:$0 \leq \pdim(\cG) \leq n$
\form#392:$x_k$
\form#393:$\mathrm{expr} = \langle \vect{a}, \vect{x} \rangle + b$
\form#394:$\bigl\{ (0, 0)^\transpose, (0, 3)^\transpose, (3, 0)^\transpose \bigr\}$
\form#395:$3x_0 + 2 x_1 + 1$
\form#396:$a_0 = 3, a_1 = 2, b = 1$
\form#397:$\cL_1$
\form#398:$\bigl\{ (1, 0)^\transpose, (7, 3)^\transpose, (10, 0)^\transpose \bigr\}$
\form#399:$(1, 0)$
\form#400:$(3, -3), (0, 9)$
\form#401:$\{x \equiv_3 1, x + y \equiv_9 1\}$
\form#402:$\cL_2$
\form#403:$x = y$
\form#404:$y$
\form#405:$\fund{\phi = (\mathrm{lhs}', \mathrm{rhs}, f)}{\Rset^n}{\Rset^n}$
\form#406:$f \in \Qset$
\form#407:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n \itc (\vect{v}, \vect{w}) \in \phi \iff \bigl( \langle \vect{c}, \vect{w} \rangle + d \equiv_f \langle \vect{a}, \vect{v} \rangle + b \bigr) \land \Bigl( \bigland_{0 \leq i < n, c_i = 0} w_i = v_i \Bigr). \]
\form#408:$\mathrm{expr} = \bigl(\langle \vect{a}, \vect{x} \rangle + b\bigr)$
\form#409:$c, f \in \Rset$
\form#410:$\cg = ( \mathrm{expr} \equiv_f c )$
\form#411:$\mathrm{expr}$
\form#412:$\mathrm{val} = \langle \vect{a}, \vect{w} \rangle + b$
\form#413:$\vect{w} \in \cL$
\form#414:\[ \lvert\mathrm{val}\rvert = \min\Bigl\{\, \bigl\lvert\langle \vect{a}, \vect{v} \rangle + b \bigr\rvert \Bigm| \vect{v} \in \cL \,\Bigr\}. \]
\form#415:$( \mathrm{expr} = c )$
\form#416:$0$
\form#417:$\cL_1, \cL_2 \in \Gset_n$
\form#418:$ \cL_1 \nearrow \cL_2$
\form#419:\[ \bigl\{\, \vect{p} + \mu \vect{q} \in \Rset^n \bigm| \vect{p} \in \cL_1, \vect{q} \in \cL_2, \mu \in \Zset \,\bigr\}. \]
\form#420:$ \cg = \bigl( \langle \vect{a}, \vect{x} \rangle \equiv_f b \bigr) $
\form#421:$\cL_{\cg} = \gcon\bigl(\{\cg\}\bigr)$
\form#422:$\cg$
\form#423:$\cL \inters \cL_{\cg} = \emptyset$
\form#424:$\cL \inters \cL_{\cg} \neq \emptyset$
\form#425:$\cL \inters \cL_{\cg} \subset \cL$
\form#426:$\cL \sseq \cL_{\cg}$
\form#427:$\cL \inters \cQ = \emptyset$
\form#428:$\cL \inters \cQ \neq \emptyset$
\form#429:$\cL \inters \cQ \subset \cL$
\form#430:$\cL \sseq \cQ$
\form#431:$=$
\form#432:$J$
\form#433:$\{0, \ldots, n-1\}$
\form#434:$w$
\form#435:$R = \{r \in \Rset \mid 0 \leq r < 2^w\}$
\form#436:$R = \{r \in \Rset \mid -2^{w-1} \leq r < 2^{w-1}\}$
\form#437:$v_j$
\form#438:$j$
\form#439:$\cL'$
\form#440:$a \in R$
\form#441:$a$
\form#442:$\cL' = \cL$
\form#443:$a + z \in R$
\form#444:$z \in \Zset$
\form#445:$(0, \ldots, 0, v_j, 0, \ldots, 0)$
\form#446:$v_j = 1$
\form#447:$v_j = a \mod 2^w$
\form#448:$a \in \Rset$
\form#449:$a'$
\form#450:$a' = a \mod 2^w$
\form#451:$a'\in R$
\form#452:$v_j = 2^w$
\form#453:$\cL_1 \widen \cL_2$
\form#454:$\cL_1 \sseq \cL_2$
\form#455:$D$
\form#456:$\mathord{\entails}$
\form#457:$\mathord{\meet}$
\form#458:$\true$
\form#459:$\false$
\form#460:$\cS \in \wp(D)$
\form#461:$\false \notin \cS$
\form#462:$\forall d_1, d_2 \in \cS \itc d_1 \entails d_2 \implies d_1 = d_2$
\form#463:$\wpfn{D}{\entails}$
\form#464:$\fund{\nonredmap}{\wpf(D)}{\wpfn{D}{\entails}}$
\form#465:$\cS \in \wpf(D)$
\form#466:\[ \nonredmap(\cS) \defeq \cS \setdiff \{\, d \in \cS \mid d = \false \text{ or } \exists d' \in \cS \st d \sentails d' \,\}. \]
\form#467:$d \sentails d'$
\form#468:$d \entails d' \land d \ne d'$
\form#469:$\cS$
\form#470:$\nonredmap(\cS)$
\form#471:$\cS \neq \{ \false \}$
\form#472:$D_{\smallP}$
\form#473:$\mathord{\entailsP}$
\form#474:$\cS_1$
\form#475:$\cS_2 \in D_{\smallP}$
\form#476:$\cS_1 \entailsP \cS_2$
\form#477:\[ \forall d_1 \in \cS_1 \itc \exists d_2 \in \cS_2 \st d_1 \entails d_2. \]
\form#478:$\{\true\}$
\form#479:$ \nonredmap \bigl( \{\, d_1 \meet d_2 \mid d_1 \in \cS_1, d_2 \in \cS_2 \,\} \bigr) $
\form#480:$\nonredmap(\cS_1 \union \cS_2)$
\form#481:$\cS \in D_{\smallP}$
\form#482:$d \in D$
\form#483:$\nonredmap\bigl(\cS \union \{d\}\bigr)$
\form#484:$\cS_1 = \{ d_1, \ldots, d_m \}$
\form#485:$\cS_2 = \{ c_1, \ldots, c_n \}$
\form#486:$\cS = \{ s_1, \ldots, s_q \}$
\form#487:$\cS_2$
\form#488:$q \leq m$
\form#489:$s_k \in \cS$
\form#490:$d_i \in \cS_1$
\form#491:$c_j \in \cS_2$
\form#492:$s_k$
\form#493:$d_i$
\form#494:$c_j$
\form#495:$\cS_1, \cS_2$
\form#496:$\uplus$
\form#497:$c, d$
\form#498:$c \uplus d \neq c \union d$
\form#499:$\cV$
\form#500:$\mathbb{F}_a$
\form#501:$\mathbb{F}_t$
\form#502:$-\infty$
\form#503:$+\infty$
\form#504:\[ i + \sum_{v \in \cV}i_{v}v, \]
\form#505:$v$
\form#506:$i_{v}$
\form#507:$\mathbb{I}_a$
\form#508:$\infty$
\form#509:$\mathbb{L}_a$
\form#510:\[ \left(i + \sum_{v \in V}i_{v}v\right) \aslf \left(i' + \sum_{v \in V}i'_{v}v\right) \defeq \left(i \asifp i'\right) + \sum_{v \in V}\left(i_{v} \asifp i'_{v}\right)v, \]
\form#511:\[ \left(i + \sum_{v \in V}i_{v}v\right) \adlf \left(i' + \sum_{v \in V}i'_{v}v\right) \defeq \left(i \adifp i'\right) + \sum_{v \in V}\left(i_{v} \adifp i'_{v}\right)v, \]
\form#512:\[ i \amlf \left(i' + \sum_{v \in V}i'_{v}v\right) \defeq \left(i \amifp i'\right) + \sum_{v \in V}\left(i \amifp i'_{v}\right)v, \]
\form#513:\[ \left(i + \sum_{v \in V}i_{v}v\right) \adivlf i' \defeq \left(i \adivifp i'\right) + \sum_{v \in V}\left(i_{v} \adivifp i'\right)v. \]
\form#514:$\asifp, \adifp, \amifp,$
\form#515:$\adivifp$
\form#516:$\fund{\rho^{\#}}{\cV}{\mathbb{I}_a}$
\form#517:$\fund{\rho^{\#}_l}{\cV}{\mathbb{L}_a}$
\form#518:$e$
\form#519:$\left \llbracket \rho^{\#}, \rho^{\#}_l \right \rrbracket$
\form#520:$\linexprenv{e}{\rho^{\#}}{\rho^{\#}_l}$
\form#521:$i+1$
\form#522:\[ \sum_{i=0}^{n-1} a_i x_i + b \]
\form#523:$x_i$
\form#524:$i + 1$
\form#525:$lf = x_5 - x_2 + 1$
\form#526:$4x - 2y - z + 14$
\form#527:$3$
\form#528:$\sum_{i=0}^{n-1} a_i x_i + b = 0$
\form#529:$\sum_{i=0}^{n-1} a_i x_i + b \geq 0$
\form#530:$\sum_{i=0}^{n-1} a_i x_i + b > 0$
\form#531:$3x + 5y - z = 0$
\form#532:$4x \geq 2y - 13$
\form#533:$2$
\form#534:$4x > 2y - 13$
\form#535:$x - 5y + 3z \leq 4$
\form#536:$x - 5y + 3z > 4$
\form#537:$\vect{l} = (a_0, \ldots, a_{n-1})^\transpose$
\form#538:$\vect{r} = (a_0, \ldots, a_{n-1})^\transpose$
\form#539:$\vect{p} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#540:$\vect{c} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#541:$d > 0$
\form#542:$x-y-z$
\form#543:$\vect{p} = (1, 0, 2)^\transpose \in \Rset^3$
\form#544:$\vect{0} \in \Rset^3$
\form#545:$\vect{0} \in \Rset^2$
\form#546:$\vect{0} \in \Rset^0$
\form#547:$\vect{p}$
\form#548:$\vect{q} = (-1.5, 3.2, 2.1)^\transpose \in \Rset^3$
\form#549:$\vect{c} = (1, 0, 2)^\transpose \in \Rset^3$
\form#550:$(a_0, \ldots, a_{n-1})^\transpose$
\form#551:$(a_0, 2 a_1, \ldots, (i+1)a_i, \ldots, n a_{n-1})^\transpose$
\form#552:$\vect{q} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#553:$\vect{p} = (1, -1, -1)^\transpose \in \Rset^3$
\form#554:$\vect{p1} = (-1.5, 3.2, 2.1)^\transpose \in \Rset^3$
\form#555:$\vect{q} = (1, 0, 2)^\transpose \in \Rset^3$
\form#556:$\cg = \sum_{i=0}^{n-1} a_i x_i + b = 0 \pmod m$
\form#557:$m \neq 0$
\form#558:$\pmod{1}$
\form#559:$4x = 2y - 13 \pmod{1}$
\form#560:$4x = 2y - 13 \pmod{2}$
\form#561:$x - 5y + 3z = 4 \pmod{5}$
\form#562:$2 e$
\form#563:$2x - 10y + 6z = 8 \pmod{5}$
\form#564:$1$
\form#565:$x + y = 1 \pmod{2}$
\form#566:$\Zset^2$
\form#567:$(0, 1)^\transpose \in \Rset^2$
\form#568:\[ \bigl\{\, (x, 0)^\transpose \in \Rset^2 \bigm| x \geq 0 \,\bigr\}, \]
\form#569:$R^0$
\form#570:$\{ 2 \} \sseq \Rset$
\form#571:\[ \bigl\{\, (2, y)^\transpose \in \Rset^2 \bigm| y \in \Rset \,\bigr\}. \]
\form#572:$\bigl\{ (2, 0)^\transpose \bigr\} \sseq \Rset^2$
\form#573:$x+4$
\form#574:$x+y$
\form#575:$x-y$
\form#576:$\bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2$
\form#577:$\bigl\{(3, 0)^\transpose \bigr\} \sseq \Rset^2$
\form#578:\[ \bigl\{\, (x, 0)^\transpose \in \Rset^2 \bigm| x \pmod{1}\ 0 \,\bigr\}, \]
\form#579:$x + y = 2$
\form#580:$x+3$
\form#581:$(x, y)$
\form#582:$x=y$
\form#583:$x=-y$
\form#584:$\pm x_i \leq k$
\form#585:$x_i - x_j \leq k$
\form#586:\[ a_i x_i - a_j x_j \relsym b \]
\form#587:$\mathord{\relsym} \in \{ \leq, =, \geq \}$
\form#588:$a_j$
\form#589:$a_i = 0$
\form#590:$a_j = 0$
\form#591:$a_i = a_j$
\form#592:$3x - 3y \leq 1$
\form#593:$x - y \leq 1$
\form#594:$x - y \leq \frac{1}{3}$
\form#595:$x - y \leq k$
\form#596:$k > \frac{1}{3}$
\form#597:$\frac{1}{3}$
\form#598:$3x - y \leq 1$
\form#599:$\Rset^3$
\form#600:\[ ax_i + bx_j \leq k \]
\form#601:$a, b \in \{-1, 0, 1\}$
\form#602:\[ \pm a_i x_i \pm a_j x_j \relsym b \]
\form#603:$3x + 3y \leq 1$
\form#604:$x + y \leq 1$
\form#605:$x + y \leq \frac{1}{3}$
\form#606:$x + y \leq k$
\form#607:$\perp$
\form#608:$M$
\form#609:$M-x'_i$
\form#610:$x'$
\form#611:$x'_i = M-x_i$
\form#612:$x_i = M-x'_i$
\form#613:$(x,y)$
\form#614:\[\left\{\begin{array}{l} y \geq 2x - 4\\ y \leq -x + p \end{array}\right.\]
\form#615:\[\left\{\begin{array}{l} M - y \geq 2M - 2x - 4\\ M - y \leq -M + x + p \end{array}\right.\]
\form#616:\[ \left\{\begin{array}{l} x' = M - \left\lfloor\frac{p+1}{3}\right\rfloor - 1 \\ y' = M - p + \left\lfloor\frac{p+1}{3}\right\rfloor + 1 \end{array}\right. \]
\form#617:\[ \left\{\begin{array}{l} x = \left\lfloor\frac{p+1}{3}\right\rfloor + 1 \\ y = p - \left\lfloor\frac{p+1}{3}\right\rfloor - 1 \end{array}\right. \]
\form#618:$x'_i-M$
\form#619:$x'_i = x_i+M$
\form#620:$x_i = x'_i-M$
\form#621:\[\left\{\begin{array}{l} y \geq -2x - 4\\ 2y \leq x + 2p \end{array}\right.\]
\form#622:\[\left\{\begin{array}{l} y' - M \geq -2x' + 2M - 4\\ 2y' - 2M \leq x' - M + 2p \end{array}\right.\]
\form#623:\[ \left\{\begin{array}{l} x' = M - \left\lfloor\frac{2p+3}{5}\right\rfloor - 1 \\ y' = M + 2\left\lfloor\frac{2p+3}{5}\right\rfloor - 2 \end{array}\right. \]
\form#624:\[ \left\{\begin{array}{l} x = -\left\lfloor\frac{2p+3}{5}\right\rfloor - 1 \\ y = 2\left\lfloor\frac{2p+3}{5}\right\rfloor - 2 \end{array}\right. \]
\form#625:$p^+-p^-$
\form#626:$p^+$
\form#627:$p^-$
\form#628:$p_i$
\form#629:$p^+_i-p^-$
\form#630:$-p^-$
\form#631:$x_1, x_2, \dots, x_n$
\form#632:$p_1, p_2, \dots, p_m$
\form#633:$f(x_2, \dots, x_n, p_1, \dots, p_m)$
\form#634:$x_1 \geq f(x_2, \dots, x_n, p_1, \dots, p_m)$
\form#635:$D_1$
\form#636:$D_2$
\form#637:$\fund{\gamma_1}{D_1}{\Rset^n}$
\form#638:$\fund{\gamma_2}{D_2}{\Rset^n}$
\form#639:$D = D_1 \times D_2$
\form#640:$\fund{\gamma}{D}{\Rset^n}$
\form#641:$d = (d_1, d_2) \in D$
\form#642:\[ \gamma(d) = \gamma_1(d_1) \inters \gamma_2(d_2). \]
\form#643:$d = (G, P) \in (\Gset \times \Pset)$
\form#644:$G$
\form#645:$[-\omega, \omega]$
\form#646:$\omega$
\form#647:$u \cdot 2^s$
\form#648:$u$
\form#649:$s$
\form#650:$\varepsilon_{\mathbf{f}}(l)$
\form#651:\[ \varepsilon_{\mathbf{f}}\left([a, b]+\sum_{v \in \cV}[a_{v}, b_{v}]v\right) \defeq (\textrm{max}(|a|, |b|) \amifp [-\beta^{-\textrm{p}}, \beta^{-\textrm{p}}]) + \sum_{v \in \cV}(\textrm{max}(|a_{v}|,|b_{v}|) \amifp [-\beta^{-\textrm{p}}, \beta^{-\textrm{p}}])v \]
\form#652:$\mathbf{f}$
\form#653:$\iota(lf)\rho^{\#}$
\form#654:\[ \iota\left(i + \sum_{v \in \cV}i_{v}v\right)\rho^{\#} \defeq i \asifp \left(\bigoplus_{v \in \cV}{}^{\#}i_{v} \amifp \rho^{\#}(v)\right) \]
\form#655:$\rho^{\#}(v)$
\form#656:$ y \cdot 2^\mathtt{exp} $
\form#657:$ y / 2^\mathtt{exp} $
\form#658:$15 \cdot 10^2 = 1500$
\form#659:$5$
\form#660:$3/580000$
\form#661:$713/10$
\form#662:$-1929/15625$
\form#663:$11/50$
\form#664:$-220001/100$
\form#665:$35$
\form#666:$44027$
\form#667:$7/2$
\form#668:$2000$
\form#669:$1024$
\form#670:$1088$
\form#671:$9073863231288$
\form#672:$17/8$
\form#673:$482$
\form#674:$256$
\form#675:\[ \{\, a \in \mathtt{to} \mid \exists b \in \mathtt{x} \st a \mathrel{\mathtt{rel}} b \,\}. \]
\form#676:\[ \{\, a \in \mathtt{to} \mid \forall b \in \mathtt{x} \itc a \mathrel{\mathtt{rel}} b \,\}. \]
\form#677:$ \{ x_1 \mapsto x_2, x_2 \mapsto x_3, x_3 \mapsto x_1 \}$
\form#678:$ x_1, x_2, x_3 $
\form#679:$O(n)$
\form#680:$O(1)$
\form#681:$O(n*\log^2 n)$
\form#682:$O(\log n)$
\form#683:$O(\log^2 n)$
\form#684:$O(k+\log^2 n)$
\form#685:$O(k+\log n)$
\form#686:$O(\log(last - first + 1))$
\form#687:$O(k)$
\form#688:$O(k*\log^2 n)$
\form#689:$O(k + \log m)$
\form#690:$0<=0$
\form#691:$0 \leq 1$
\form#692:$\sum_{i=0}^{n-1} 0 x_i + 0 = 0$
\form#693:$\sum_{i=0}^{n-1} 0 x_i + b \geq 0$
\form#694:$b \geq 0$
\form#695:$\sum_{i=0}^{n-1} 0 x_i + b > 0$
\form#696:$b > 0$
\form#697:$\sum_{i=0}^{n-1} 0 x_i + b = 0$
\form#698:$b \neq 0$
\form#699:$b < 0$
\form#700:$b \leq 0$
\form#701:$\epsilon \geq 0$
\form#702:$\epsilon \leq 1$
\form#703:$0<0$
\form#704:$L_\infty$
\form#705:$ e_1 = e_2 \pmod{m}$
\form#706:$ e_1 = e_2 \pmod{mk}$
\form#707:$\sum_{i=0}^{n-1} 0 x_i + 0 == 0$
\form#708:$\sum_{i=0}^{n-1} 0 x_i + b \%= 0 / m$
\form#709:$b = 0 \pmod{m}$
\form#710:$\sum_{i=0}^{n-1} 0 x_i + b == 0$
\form#711:$b \neq 0 \pmod{m}$
\form#712:$0 = 1 \pmod{1}$
\form#713:$0 = 1 \pmod{0}$
\form#714:$e1 = e2 \pmod{1}$
\form#715:$e = n \pmod{1}$
\form#716:$n = e \pmod{1}$
\form#717:$\mathrm{var}' \relsym \frac{\mathrm{expr}}{\mathrm{denominator}}$
\form#718:$\mathrm{lhs}' \relsym \mathrm{rhs}$
\form#719:$\frac{\mathrm{lb\_expr}}{\mathrm{denominator}} \leq \mathrm{var}' \leq \frac{\mathrm{ub\_expr}}{\mathrm{denominator}}$
\form#720:$\mathtt{y} \Delta \mathtt{x}$
\form#721:$\cB \sseq \Rset^2$
\form#722:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cB \,\bigr\}. \]
\form#723:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cB \,\bigr\}. \]
\form#724:$B \sseq \Rset^n$
\form#725:$D \sseq \Rset^m$
\form#726:$R \sseq \Rset^{n+m}$
\form#727:\[ R \defeq \Bigl\{\, (x_1, \ldots, x_n, y_1, \ldots, y_m)^\transpose \Bigm| (x_1, \ldots, x_n)^\transpose \in B, (y_1, \ldots, y_m)^\transpose \in D \,\Bigl\}. \]
\form#728:$f(k)$
\form#729:$\dots$
\form#730:$I$
\form#731:$d$
\form#732:$n/d$
\form#733:$\times$
\form#734:$\epsilon$
\form#735:$M \in \Rset^r \times \Rset^c$
\form#736:$N \in \Rset^{r+n} \times \Rset^{c+n}$
\form#737:$N = \bigl(\genfrac{}{}{0pt}{}{0}{M}\genfrac{}{}{0pt}{}{J}{o}\bigr)$
\form#738:$n \times n$
\form#739:$\sum_{i = 0}^{n - 1} a_i x_i + b$
\form#740:${a'}_{ij}$
\form#741:$a_{ij}$
\form#742:\[ {a'}_{ij} = \begin{cases} a_{ij} * \mathrm{denominator} + a_{iv} * \mathrm{expr}[j] \quad \text{for } j \neq v; \\ \mathrm{expr}[v] * a_{iv} \quad \text{for } j = v. \end{cases} \]
\form#743:$ e_1 = e_2 $
\form#744:$ e_1 = e_2 \pmod{0}$
\form#745:$r \times c$
\form#746:$(r+dims) \times (c+dims)$
\form#747:$\bigl(\genfrac{}{}{0pt}{}{0}{A} \genfrac{}{}{0pt}{}{B}{A}\bigr)$
\form#748:$B$
\form#749:$dims \times dims$
\form#750:$\bigl(\genfrac{}{}{0pt}{}{0}{1} \genfrac{}{}{0pt}{}{1}{0}\bigr)$
\form#751:\[ \frac{\sum_{i = 0}^{n - 1} a_i x_i + b} {\mathrm{denominator}}. \]
\form#752:$\leq$
\form#753:$\relsym$
\form#754:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cP \,\bigr\}. \]
\form#755:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cP \,\bigr\}. \]
\form#756:$\Rset^0 = \{\cdot\}$
\form#757:\[ \begin{cases} sat\_c[i][j] = 0, \quad \text{if } G[i] \cdot C^\mathrm{T}[j] = 0; \\ sat\_c[i][j] = 1, \quad \text{if } G[i] \cdot C^\mathrm{T}[j] > 0. \end{cases} \]
\form#758:\[ \begin{cases} sat\_g[i][j] = 0, \quad \text{if } C[i] \cdot G^\mathrm{T}[j] = 0; \\ sat\_g[i][j] = 1, \quad \text{if } C[i] \cdot G^\mathrm{T}[j] > 0. \end{cases} \]
\form#759:$O(r)$
\form#760:$O(n+k)$
\form#761:$O(\sum_{j=1}^{r} (k_j*\log^2 n_j))$
\form#762:$k_j$
\form#763:$n_j$
\form#764:$O(r+k*\log^2 c)$
\form#765:$(r+n) \times (c+m)$
\form#766:$\bigl(\genfrac{}{}{0pt}{}{M}{0} \genfrac{}{}{0pt}{}{0}{0}\bigr)$
\form#767:$(r+n) \times c$
\form#768:$\genfrac{(}{)}{0pt}{}{M}{0}$
\form#769:$(r+1) \times c$
\form#770:$\genfrac{(}{)}{0pt}{}{M}{y}$
\form#771:$ \{ 1 \mapsto 3, 2 \mapsto 4, 3 \mapsto 6, 4 \mapsto 2, 5 \mapsto 5, 6 \mapsto 1 \}$
\form#772:$(1 3 6)(2 4)$
\form#773:$O(k*\sum_{j=1}^{r} \log^2 n_j)$
\form#774:$O(k*r*\log^2 c)$
\form#775:$r \times (c+n)$
\form#776:$(M \, 0)$
\form#777:$O(\sum_{j=1}^{r} (k_j+\log n_j))$
\form#778:$O(k+r*\log c)$
\form#779:$O(k + \sum_{j=1}^{r} (\log^2 n_j))$
\form#780:$O(r*(c-i+\log^2 c))$
\form#781:$O(\sum_{j=1}^r (k_j*\log n_j))$
\form#782:$O(r*n*\log c)$
\form#783:$O(n*\log n)$
\form#784:$O(r+k)$
\form#785:$[\mathrm{first}, \mathrm{last})$
\form#786:$\frac{\mathtt{numer}}{\mathtt{denom}}$
\form#787:$\frac{d_{j}}{\|\Delta x^{j} \|}$
\form#788:\[ \|\Delta x^{j} \| = \left( 1+\sum_{i=1}^{m} \alpha_{ij}^2 \right)^{\frac{1}{2}}. \]
\form#789:$\alpha$
\form#790:$\rho^{\#}$
\form#791:$\rho^{\#}_l$
\form#792:\[ \langle \vect{c}, \vect{r}_1 \rangle = 0 \land \langle \vect{c}, \vect{r}_2 \rangle = 0 \Rightarrow \langle \vect{c}, (\lambda_1 \vect{r}_1 + \lambda_2 \vect{r}_2) \rangle = \lambda_1 \langle \vect{c}, \vect{r}_1 \rangle + \lambda_2 \langle \vect{c}, \vect{r}_2 \rangle = 0, \]
\form#793:$\lambda_1, \lambda_2$
\form#794:$\bigl(\genfrac{}{}{0pt}{}{A}{0} \genfrac{}{}{0pt}{}{0}{B}\bigr)$
\form#795:$\bigl(\genfrac{}{}{0pt}{}{1}{0} \genfrac{}{}{0pt}{}{0}{1}\bigr)$
\form#796:$\mathrm{var}' = \frac{\mathrm{expr}}{\mathrm{denominator}} \pmod{\mathrm{modulus}}$
\form#797:$\mathrm{lhs}' = \mathrm{rhs} \pmod{\mathrm{modulus}}$
\form#798:$\cL \sseq \Rset^2$
\form#799:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cL \,\bigr\}. \]
\form#800:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cL \,\bigr\}. \]
\form#801:$\{3x \equiv_3 0, 4x + y \equiv_3 1\}$
\form#802:$\{3x \equiv_3 0, x + y \equiv_3 1\}$
\form#803:$\mathrm{var} \relsym \frac{\mathrm{expr}}{\mathrm{denominator}}$
\form#804:$l = c$
\form#805:$l = ax + c$
\form#806:$l = ax + by + c$
\form#807:$ax + b = cy + d$
\form#808:$\cO \sseq \Rset^2$
\form#809:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cO \,\bigr\}. \]
\form#810:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cO \,\bigr\}. \]
\form#811:$O(n^2)$
\form#812:$i + \sum_{v \in \cV}i_{v}v $
\form#813:$i' + \sum_{v \in \cV}i'_{v}v $
\form#814:$\aslf$
\form#815:\[ \left(i + \sum_{v \in \cV}i_{v}v \right) \aslf \left(i' + \sum_{v \in \cV}i'_{v}v \right) = \left(i \asifp i'\right) + \sum_{v \in \cV}\left(i_{v} \asifp i'_{v} \right)v. \]
\form#816:$e_{1} \oplus e_{2}$
\form#817:$\linexprenv{e_{1} \oplus e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\form#818:\[ \linexprenv{e_{1} \oplus e_{2}}{\rho^{\#}}{\rho^{\#}_l} = \linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \aslf \linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l} \aslf \varepsilon_{\mathbf{f}}\left(\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \right) \aslf \varepsilon_{\mathbf{f}}\left(\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l} \right) \aslf mf_{\mathbf{f}}[-1, 1] \]
\form#819:$l$
\form#820:$mf_{\mathbf{f}}$
\form#821:$\adlf$
\form#822:\[ \left(i + \sum_{v \in \cV}i_{v}v\right) \aslf \left(i' + \sum_{v \in \cV}i'_{v}v\right) = \left(i \asifp i'\right) + \sum_{v \in \cV}\left(i_{v} \asifp i'_{v}\right)v, \]
\form#823:\[ \left(i + \sum_{v \in \cV}i_{v}v\right) \adlf \left(i' + \sum_{v \in \cV}i'_{v}v\right) = \left(i \adifp i'\right) + \sum_{v \in \cV}\left(i_{v} \adifp i'_{v}\right)v. \]
\form#824:$e_{1} \ominus e_{2}$
\form#825:$\linexprenv{e_{1} \ominus e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\form#826:\[ \linexprenv{e_{1} \ominus e_{2}}{\rho^{\#}}{\rho^{\#}_l} = \linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \adlf \linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l} \aslf \varepsilon_{\mathbf{f}}\left(\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \right) \aslf \varepsilon_{\mathbf{f}}\left(\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l} \right) \aslf mf_{\mathbf{f}}[-1, 1] \]
\form#827:$\amlf$
\form#828:\[ i \amlf \left(i' + \sum_{v \in \cV}i'_{v}v\right) = \left(i \amifp i'\right) + \sum_{v \in \cV}\left(i \amifp i'_{v}\right)v. \]
\form#829:$[a, b] \otimes e_{2}$
\form#830:$\linexprenv{[a, b] \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\form#831:\[ \linexprenv{[a, b] \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l} = \left([a, b] \amlf \linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l}\right) \aslf \left([a, b] \amlf \varepsilon_{\mathbf{f}}\left(\linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l} \right)\right) \aslf mf_{\mathbf{f}}[-1, 1]. \]
\form#832:$e_{1} \otimes [a, b]$
\form#833:$\linexprenv{e_{1} \otimes [a, b]}{\rho^{\#}}{\rho^{\#}_l}$
\form#834:\[ \linexprenv{e_{1} \otimes [a, b]}{\rho^{\#}}{\rho^{\#}_l} = \linexprenv{[a, b] \otimes e_{1}}{\rho^{\#}}{\rho^{\#}_l}. \]
\form#835:$e_{1} \otimes e_{2}$
\form#836:$\linexprenv{e_{1} \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\form#837:\[ \linexprenv{e_{1} \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l} = \linexprenv{\iota\left(\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \right)\rho^{\#} \otimes e_{2}}{\rho^{\#}}{\rho^{\#}_l}, \]
\form#838:$\iota(l)\rho^{\#}$
\form#839:$\adivlf$
\form#840:\[ \left(i + \sum_{v \in \cV}i_{v}v\right) \adivlf i' = \left(i \adivifp i'\right) + \sum_{v \in \cV}\left(i_{v} \adivifp i'\right)v. \]
\form#841:$e_{1} \oslash [a, b]$
\form#842:$ \linexprenv{e_{1} \oslash [a, b]}{\rho^{\#}}{\rho^{\#}_l} $
\form#843:\[ \linexprenv{e_{1} \oslash [a, b]}{\rho^{\#}}{\rho^{\#}_l} = \left(\linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \adivlf [a, b]\right) \aslf \left(\varepsilon_{\mathbf{f}}\left( \linexprenv{e_{1}}{\rho^{\#}}{\rho^{\#}_l} \right) \adivlf [a, b]\right) \aslf mf_{\mathbf{f}}[-1, 1], \]
\form#844:$e_{1} \oslash e_{2}$
\form#845:$\linexprenv{e_{1} \oslash e_{2}}{\rho^{\#}}{\rho^{\#}_l}$
\form#846:\[ \linexprenv{e_{1} \oslash e_{2}}{\rho^{\#}}{\rho^{\#}_l} = \linexprenv{e_{1} \oslash \iota\left( \linexprenv{e_{2}}{\rho^{\#}}{\rho^{\#}_l} \right)\rho^{\#}}{\rho^{\#}}{\rho^{\#}_l}, \]
\form#847:$\frac{\mathtt{expr}}{\mathtt{d}}$
\form#848:$f_i(x,p) = 0 ; 1 \leq i \leq n$
\form#849:$f_i(x,p) \geq 0 ; 1 \leq i \leq n$
\form#850:$\sum\limits_{i=1}^n f_i(x,p) \leq 0$
\form#851:$c=s_{*j}\frac{t_{ik}}{s_{ij}}$
\form#852:$c'=s_{*j'}\frac{t_{i'k}}{s_{i'j'}}$
\form#853:$s_{*j}$
\form#854:$s_{*j'}$
\form#855:$j'$
\form#856:$(i,j)$
\form#857:$c'$
\form#858:$(i',j')$
\form#859:$-c$
\form#860:$-c'$
\form#861:\[ \mathit{ps} = \bigl\{ \{ x \geq 0 \}, \{ x \leq 0 \} \bigr\}, \]
\form#862:$\mathit{ps}$
\form#863:$\cP \uplus \cQ \neq \cP \union \cQ$
\form#864:$ x'_1, \ldots, x'_n $
\form#865:$ 0, \ldots, n-1 $
\form#866:$ x_1, \ldots, x_n $
\form#867:$ n, \ldots, 2n-1 $
\form#868:$ n+1 $
\form#869:$ \mu_0 + \sum_{i=1}^n \mu_i x_i $
\form#870:$ \mu_0, \mu_1, \ldots, \mu_n $
\form#871:$ n, 0, \ldots, n-1 $
\form#872:$\linexprenv{cast(e)}{\rho^{\#}}{\rho^{\#}_l}$
\form#873:\[ \linexprenv{cast(e)}{\rho^{\#}}{\rho^{\#}_l} = \linexprenv{e}{\rho^{\#}}{\rho^{\#}_l} \aslf \varepsilon_{\mathbf{f}}\left(\linexprenv{e}{\rho^{\#}}{\rho^{\#}_l} \right) \aslf mf_{\mathbf{f}}[-1, 1] \]
\form#874:$[l, u]$
\form#875:$\linexprenv{v}{\rho^{\#}}{\rho^{\#}_l}$
\form#876:$\rho^{\#}_l(v)$
\form#877:$[-1, 1]v$
\form#878:$\oplus$
\form#879:$\ominus$
\form#880:$\otimes$
\form#881:$\oslash$
\form#882:\[ \adlf \left(i + \sum_{v \in \cV}i_{v}v\right) = \left(\adifp i\right) + \sum_{v \in \cV}\left(\adifp i_{v} \right)v, \]
\form#883:$\ominus e$
\form#884:$\linexprenv{\ominus e}{\rho^{\#}}{\rho^{\#}_l}$
\form#885:\[ \linexprenv{\ominus e}{\rho^{\#}}{\rho^{\#}_l} = \adlf \left( \linexprenv{e}{\rho^{\#}}{\rho^{\#}_l} \right). \]
