\form#0:$n$
\form#1:$\Rset^n$
\form#2:$\Rset$
\form#3:$\nonnegRset$
\form#4:$i \in \{0, \ldots, n-1\}$
\form#5:$v_i$
\form#6:$i$
\form#7:$\vect{v} = (v_0, \ldots, v_{n-1})^\transpose \in \Rset^n$
\form#8:$\vect{0}$
\form#9:$\vect{v} \in \Rset^n$
\form#10:$\Rset^{n \times 1}$
\form#11:$\vect{v}^\transpose$
\form#12:$\vect{v},\vect{w} \in \Rset^n$
\form#13:$\langle \vect{v}, \vect{w} \rangle$
\form#14:\[ \vect{v}^\transpose \vect{w} = \sum_{i=0}^{n-1} v_i w_i. \]
\form#15:$S_1, S_2 \sseq \Rset^n$
\form#16:$S_1$
\form#17:$S_2$
\form#18:$S_1 + S_2 = \{\, \vect{v}_1 + \vect{v}_2 \mid \vect{v}_1 \in S_1, \vect{v}_2 \in S_2 \,\}.$
\form#19:$\vect{a} \in \Rset^n$
\form#20:$b \in \Rset$
\form#21:$\vect{a} \neq \vect{0}$
\form#22:$\mathord{\relsym} \in \{ =, \geq, > \}$
\form#23:$\langle \vect{a}, \vect{x} \rangle \relsym b$
\form#24:$\mathord{\relsym} \in \{ = \}$
\form#25:$\mathord{\relsym} \in \{ \geq \}$
\form#26:$\mathord{\relsym} \in \{ > \}$
\form#27:$\langle \vect{a}, \vect{x} \rangle = b$
\form#28:$\langle \vect{a}, \vect{x} \rangle \geq b$
\form#29:$\langle -\vect{a}, \vect{x} \rangle \geq -b$
\form#30:$\vect{a} = \vect{0}$
\form#31:$\langle \vect{0}, \vect{x} \rangle \relsym b$
\form#32:$\emptyset$
\form#33:$\cP \sseq \Rset^n$
\form#34:$\cP$
\form#35:$n = 0$
\form#36:$\cP = \emptyset$
\form#37:$\Pset_n$
\form#38:$\cP \in \Pset_n$
\form#39:$\CPset_n$
\form#40:$\lambda \in \nonnegRset$
\form#41:\[ \cP \sseq \bigl\{\, \vect{x} \in \Rset^n \bigm| - \lambda \leq x_j \leq \lambda \text{ for } j = 0, \ldots, n-1 \,\bigr\}. \]
\form#42:\[ \cP \defeq \{\, \vect{x} \in \Rset^n \mid A_1 \vect{x} = \vect{b}_1, A_2 \vect{x} \geq \vect{b}_2, A_3 \vect{x} > \vect{b}_3 \,\}, \]
\form#43:$i \in \{1, 2, 3\}$
\form#44:$A_i \in \Rset^{m_i} \times \Rset^n$
\form#45:$\vect{b}_i \in \Rset^{m_i}$
\form#46:$m_1, m_2, m_3 \in \Nset$
\form#47:$S = \{ \vect{x}_1, \ldots, \vect{x}_k \} \sseq \Rset^n$
\form#48:$\lambda_1, \ldots, \lambda_k \in \Rset$
\form#49:$\vect{v} = \sum_{j=1}^k \lambda_j \vect{x}_j$
\form#50:$S$
\form#51:$\forall j \in \{ 1, \ldots, k \} \itc \lambda_j \in \nonnegRset$
\form#52:$\sum_{j = 1}^k \lambda_j = 1$
\form#53:$\linearhull(S)$
\form#54:$\conichull(S)$
\form#55:$\affinehull(S)$
\form#56:$\convexhull(S)$
\form#57:$P, C \sseq \Rset^n$
\form#58:$P \union C = S$
\form#59:$\NNChull(P, C)$
\form#60:$\lambda_j > 0$
\form#61:$\vect{x}_j \in P$
\form#62:$P$
\form#63:$\NNChull(P, C) = \NNChull(P, P \union C)$
\form#64:$C \sseq P$
\form#65:\[ \convexhull(P) = \NNChull(P, \emptyset) = \NNChull(P, P) = \NNChull(P, C). \]
\form#66:$\vect{p} \in \cP$
\form#67:$\vect{c} \in \Rset^n$
\form#68:$\vect{r} \in \Rset^n$
\form#69:$\vect{r} \neq \vect{0}$
\form#70:$\cP \neq \emptyset$
\form#71:$\vect{p} + \lambda \vect{r} \in \cP$
\form#72:$\vect{l} \in \Rset^n$
\form#73:$\vect{l}$
\form#74:$-\vect{l}$
\form#75:$\vect{r}$
\form#76:$\vect{r}_1$
\form#77:$\vect{r}_2$
\form#78:$\vect{r} \neq \lambda \vect{r}_1$
\form#79:$\vect{r} \neq \lambda \vect{r}_2$
\form#80:$\vect{r}_1 \neq \lambda \vect{r}_2$
\form#81:$L$
\form#82:$R$
\form#83:$C$
\form#84:$\cG = (L, R, P, C)$
\form#85:\[ \cP = \linearhull(L) + \conichull(R) + \NNChull(P, C), \]
\form#86:$+$
\form#87:$\cP \in \CPset_n$
\form#88:$\cG = (L, R, P)$
\form#89:\[ \cP = \linearhull(L) + \conichull(R) + \convexhull(P). \]
\form#90:$P = \emptyset$
\form#91:$\Rset^2$
\form#92:$y \geq 0$
\form#93:$L = \bigl\{ (1, 0)^\transpose \bigr\}$
\form#94:$R = \bigl\{ (0, 1)^\transpose \bigr\}$
\form#95:$P = \bigl\{ (0, 0)^\transpose \bigr\}$
\form#96:$C = \emptyset$
\form#97:$\cC$
\form#98:$\cG' = (L', R', P', C') \neq \cG$
\form#99:$L' \sseq L$
\form#100:$R' \sseq R$
\form#101:$P' \sseq P$
\form#102:$C' \sseq C$
\form#103:$\cG$
\form#104:$(\cC, \cG)$
\form#105:$(L, R, P)$
\form#106:$n \in \Nset$
\form#107:$\vect{a}, \vect{x} \in \Rset^m$
\form#108:$m \leq n$
\form#109:$\vect{x} \in \Rset^m$
\form#110:$\{ \vect{x}_1, \ldots, \vect{x}_k \} \sseq \Rset^n$
\form#111:\[ \sum_{i = 1}^k \lambda_i \vect{x}_i = \vect{0}, \quad \sum_{i = 1}^k \lambda_i = 0 \]
\form#112:$i = 1, \ldots, k$
\form#113:$\lambda_i = 0$
\form#114:$n + 1$
\form#115:$k \in \Nset$
\form#116:$\pdim(\cP) = k$
\form#117:$k + 1$
\form#118:$0 \leq \pdim(\cP) \leq n$
\form#119:$-1$
\form#120:$k \leq n$
\form#121:$\pdim(\cP) \neq \pdim(\cQ)$
\form#122:$\cQ$
\form#123:$\pdim(\cP) = \pdim(\cQ)$
\form#124:$\cP_1, \cP_2 \in \Pset_n$
\form#125:$\cP_1$
\form#126:$\cP_2$
\form#127:$\cP_1 \inters \cP_2$
\form#128:$\cP_1 \uplus \cP_2$
\form#129:$\cP_1, \cP_2 \in \CPset_n$
\form#130:$\cQ \in \Pset_m$
\form#131:$\cR \in \Pset_{n+m}$
\form#132:\[ \cR \defeq \Bigl\{\, (x_0, \ldots, x_{n-1}, y_0, \ldots, y_{m-1})^\transpose \in \Rset^{n+m} \Bigm| (x_0, \ldots, x_{n-1})^\transpose \in \cP, (y_0, \ldots, y_{m-1})^\transpose \in \cQ \,\Bigl\}. \]
\form#133:$n+m$
\form#134:$\cQ \in \Pset_{n+i}$
\form#135:$i+n$
\form#136:$\cP \sseq \Rset^2$
\form#137:\[ \cQ = \bigl\{\, (x_0, x_1, x_2)^\transpose \in \Rset^3 \bigm| (x_0, x_1)^\transpose \in \cP \,\bigr\}. \]
\form#138:\[ \cQ = \bigl\{\, (x_0, x_1, 0)^\transpose \in \Rset^3 \bigm| (x_0, x_1)^\transpose \in \cP \,\bigr\}. \]
\form#139:$\cP \in \Pset_4$
\form#140:$\bigl\{ (3, 1, 0, 2)^\transpose \bigr\} \sseq \Rset^4$
\form#141:$\{x_1, x_2\}$
\form#142:\[ \cQ = \bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2. \]
\form#143:$m$
\form#144:$m = 2$
\form#145:\[ \cQ = \bigl\{ (3, 1)^\transpose \bigr\} \sseq \Rset^2. \]
\form#146:$\pard{\rho}{\{0, \ldots, n-1\}}{\Nset}$
\form#147:$\rho\bigl(\{0, \ldots, n-1\}\bigr) = \{0, \ldots, m-1\}$
\form#148:$\rho$
\form#149:$m = 0$
\form#150:$\Rset^0$
\form#151:\[ \cQ \defeq \Bigl\{\, \bigl(v_{\rho^{-1}(0)}, \ldots, v_{\rho^{-1}(m-1)}\bigr)^\transpose \Bigm| (v_0, \ldots, v_{n-1})^\transpose \in \cP \,\Bigr\}. \]
\form#152:$n > 0$
\form#153:$n+1$
\form#154:$\ldots$
\form#155:$n+m-1$
\form#156:\[ \cQ \defeq \sset{ \vect{u} \in \Rset^{n+m} }{ \exists \vect{v}, \vect{w} \in \cP \st u_i = v_i \\ \qquad \mathord{} \land \forall j = n, n+1, \ldots, n+m-1 \itc u_j = w_i \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_k = v_k = w_k }. \]
\form#157:$J = \{ j_0, \ldots, j_{m-1} \}$
\form#158:$m < n$
\form#159:$j < n$
\form#160:$j \in J$
\form#161:$i < n$
\form#162:$i \notin J$
\form#163:\[ \cQ \defeq \biguplus_{d = 0}^m \cQ_d \]
\form#164:\[ \cQ_m \defeq \sset{ \vect{u} \in \Rset^{n-m} }{ \exists \vect{v} \in \cP \st u_{i'} = v_i \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_{k'} = v_k } \]
\form#165:$ d = 0 $
\form#166:$ \ldots $
\form#167:$ m-1 $
\form#168:\[ \cQ_d \defeq \sset{ \vect{u} \in \Rset^{n-m} }{ \exists \vect{v} \in \cP \st u_{i'} = v_{j_d} \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_{k'} = v_k }, \]
\form#169:$ k = 0 $
\form#170:$ n-1 $
\form#171:\[ k' \defeq k - \card \{\, j \in J \mid k > j \,\}, \]
\form#172:$\card S$
\form#173:$\reld{\phi}{\Rset^n}{\Rset^m}$
\form#174:$\phi(S) \sseq \Rset^m$
\form#175:$\phi$
\form#176:$S \sseq \Rset^n$
\form#177:\[ \phi(S) \defeq \bigl\{\, \vect{w} \in \Rset^m \bigm| \exists \vect{v} \in S \st (\vect{v}, \vect{w}) \in \phi \,\bigr\}. \]
\form#178:$\phi^{-1}(S') \sseq \Rset^n$
\form#179:$S' \sseq \Rset^m$
\form#180:\[ \phi^{-1}(S') \defeq \bigl\{\, \vect{v} \in \Rset^n \bigm| \exists \vect{w} \in S' \st (\vect{v}, \vect{w}) \in \phi \,\bigr\}. \]
\form#181:$n = m$
\form#182:$\ell \in \Nset$
\form#183:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^m \itc (\vect{v}, \vect{w}) \in \phi \iff \bigland_{i=1}^{\ell} \bigl( \langle \vect{c}_i, \vect{w} \rangle \relsym_i \langle \vect{a}_i, \vect{v} \rangle + b_i \bigr), \]
\form#184:$\vect{a}_i \in \Rset^n$
\form#185:$\vect{c}_i \in \Rset^m$
\form#186:$b_i \in \Rset$
\form#187:$\mathord{\relsym}_i \in \{ <, \leq, =, \geq, > \}$
\form#188:$i = 1, \ldots, \ell$
\form#189:$A \in \Rset^m \times \Rset^n$
\form#190:$\vect{b} \in \Rset^m$
\form#191:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^m \itc (\vect{v}, \vect{w}) \in \phi \iff \vect{w} = A\vect{v} + \vect{b}. \]
\form#192:$<$
\form#193:$>$
\form#194:$\reld{\phi}{\Rset^n}{\Rset^n}$
\form#195:$\vect{x} = (x_0, \ldots, x_{n-1})^\transpose$
\form#196:$\vect{x}' = (x'_0, \ldots, x'_{n-1})^\transpose$
\form#197:$x'_i$
\form#198:$x'_i = x_i$
\form#199:$\reld{\phi}{\Rset^3}{\Rset^3}$
\form#200:$x'_0 - x'_2 \geq 2 x_0 - x_1$
\form#201:$x'_1$
\form#202:\[ \forall \vect{v} \in \Rset^3, \vect{w} \in \Rset^3 \itc (\vect{v}, \vect{w}) \in \phi \iff (w_0 + w_2 \geq 2 v_0 - v_1) \land (w_1 = v_1). \]
\form#203:$x'_0 + 0 \cdot x'_1 - x'_2 \geq 2 x_0 - x_1$
\form#204:$x'_k$
\form#205:$\langle \vect{a}, \vect{x} \rangle + b$
\form#206:$\fund{\phi = \bigl(x'_k = \langle \vect{a}, \vect{x} \rangle + b\bigr)} {\Rset^n}{\Rset^n}$
\form#207:\[ \forall \vect{v} \in \Rset^n \itc \phi(\vect{v}) = A\vect{v} + \vect{b}, \]
\form#208:\[ A = \begin{pmatrix} 1 & & 0 & 0 & \cdots & \cdots & 0 \\ & \ddots & & \vdots & & & \vdots \\ 0 & & 1 & 0 & \cdots & \cdots & 0 \\ a_0 & \cdots & a_{k-1} & a_k & a_{k+1} & \cdots & a_{n-1} \\ 0 & \cdots & \cdots & 0 & 1 & & 0 \\ \vdots & & & \vdots & & \ddots & \\ 0 & \cdots & \cdots & 0 & 0 & & 1 \end{pmatrix}, \qquad \vect{b} = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ b \\ 0 \\ \vdots \\ 0 \end{pmatrix} \]
\form#209:$a_i$
\form#210:$b$
\form#211:$(k+1)$
\form#212:$A$
\form#213:$\vect{b}$
\form#214:$(v_0, \ldots, v_{n-1})^\transpose$
\form#215:\[ \Bigl(v_0, \ldots, \bigl(\textstyle{\sum_{i=0}^{n-1}} a_i v_i + b\bigr), \ldots, v_{n-1}\Bigr)^\transpose. \]
\form#216:$x'_k = \langle \vect{a}, \vect{x} \rangle + b$
\form#217:$\bigl\{ (0, 0)^\transpose, (0, 3)^\transpose, (3, 0)^\transpose, (3, 3)^\transpose \bigr\}$
\form#218:$x_0$
\form#219:$x_0 + 2 x_1 + 4$
\form#220:$k = 0$
\form#221:$a_0 = 1, a_1 = 2, b = 4$
\form#222:$\bigl\{ (4, 0)^\transpose, (10, 3)^\transpose, (7, 0)^\transpose, (13, 3)^\transpose \bigr\}$
\form#223:$x_0 - 2 x_1$
\form#224:$x_1$
\form#225:$a_0 = 0, a_1 = 1, b = 0$
\form#226:$(1, 0)^\transpose$
\form#227:$a_k$
\form#228:$\mathrm{lb} = \langle \vect{a}, \vect{x} \rangle + b$
\form#229:$\mathrm{ub} = \langle \vect{c}, \vect{x} \rangle + d$
\form#230:$\phi = (\mathrm{lb} \leq x'_k \leq \mathrm{ub})$
\form#231:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n \itc (\vect{v}, \vect{w}) \in \phi \iff \bigl( \langle \vect{a}, \vect{v} \rangle + b \leq w_k \leq \langle \vect{c}, \vect{v} \rangle + d \bigr) \land \Bigl( \bigland_{0 \leq i < n, i \neq k} w_i = v_i \Bigr). \]
\form#232:$\phi = (\mathrm{lhs}' \relsym \mathrm{rhs})$
\form#233:$\mathrm{lhs} = \langle \vect{c}, \vect{x} \rangle + d$
\form#234:$\mathrm{rhs} = \langle \vect{a}, \vect{x} \rangle + b$
\form#235:$\mathord{\relsym} \in \{ <, \leq, =, \geq, > \}$
\form#236:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n \itc (\vect{v}, \vect{w}) \in \phi \iff \bigl( \langle \vect{c}, \vect{w} \rangle + d \relsym \langle \vect{a}, \vect{v} \rangle + b \bigr) \land \Bigl( \bigland_{0 \leq i < n, c_i = 0} w_i = v_i \Bigr). \]
\form#237:$\mathrm{lhs} = x_k$
\form#238:$x'_k = \mathrm{rhs}$
\form#239:$\mathrm{lhs}$
\form#240:$\mathrm{rhs}$
\form#241:$\mathrm{lhs}' \leq \mathrm{rhs}$
\form#242:$\mathrm{rhs}' \geq \mathrm{lhs}$
\form#243:$\cP, \cQ \in \Pset_n$
\form#244:$ \cP \nearrow \cQ$
\form#245:\[ \bigl\{\, \vect{p} + \lambda \vect{q} \in \Rset^n \bigm| \vect{p} \in \cP, \vect{q} \in \cQ, \lambda \in \nonnegRset \,\bigr\}. \]
\form#246:$\cP,\cQ \in \CPset_n$
\form#247:$ c = \bigl( \langle \vect{a}, \vect{x} \rangle \relsym b \bigr) $
\form#248:$c$
\form#249:$\cP \inters \cQ = \emptyset$
\form#250:$\cP \inters \cQ \neq \emptyset$
\form#251:$\cP \inters \cQ \subset \cP$
\form#252:$\cP \sseq \cQ$
\form#253:$\cP \sseq \cH$
\form#254:$\cH$
\form#255:$g$
\form#256:$\cB$
\form#257:$\vect{e}_i = (0, \ldots, 1, \ldots, 0)^\transpose$
\form#258:$\langle \vect{e}_i, \vect{x} \rangle \relsym b$
\form#259:$\mathord{\relsym}$
\form#260:$\mathord{\geq}$
\form#261:$\mathord{>}$
\form#262:$\langle\vect{e}_i,\vect{x}\rangle \relsym b$
\form#263:$\mathord{\leq}$
\form#264:$\mathord{<}$
\form#265:$\cP \widen \cQ$
\form#266:$\cP, \cQ \in \CPset_n$
\form#267:$\cP \Sseq \cP \widen \cQ$
\form#268:$\cQ \sseq \cP$
\form#269:$\cP = \cQ$
\form#270:$k$
\form#271:$\mathord{\relsym} \in \{ =, \geq\}$
\form#272:$i, j \in \{ 0, \ldots, n-1 \}$
\form#273:$a_i, a_j \in \{ -1, 0, 1 \}$
\form#274:$a_i \neq a_j$
\form#275:$a_k = 0$
\form#276:$k \notin \{ i, j \}$
\form#277:$D$
\form#278:$\mathord{\entails}$
\form#279:$\mathord{\meet}$
\form#280:$\true$
\form#281:$\false$
\form#282:$\cS \in \wp(D)$
\form#283:$\false \notin \cS$
\form#284:$\forall d_1, d_2 \in \cS \itc d_1 \entails d_2 \implies d_1 = d_2$
\form#285:$\wpfn{D}{\entails}$
\form#286:$\fund{\nonredmap}{\wpf(D)}{\wpfn{D}{\entails}}$
\form#287:$\cS \in \wpf(D)$
\form#288:\[ \nonredmap(\cS) \defeq \cS \setdiff \{\, d \in \cS \mid d = \false \text{ or } \exists d' \in \cS \st d \sentails d' \,\}. \]
\form#289:$d \sentails d'$
\form#290:$d \entails d' \land d \ne d'$
\form#291:$\cS$
\form#292:$\nonredmap(\cS)$
\form#293:$\cS \neq \{ \false \}$
\form#294:$D_{\smallP}$
\form#295:$\mathord{\entailsP}$
\form#296:$\cS_1$
\form#297:$\cS_2 \in D_{\smallP}$
\form#298:$\cS_1 \entailsP \cS_2$
\form#299:\[ \forall d_1 \in \cS_1 \itc \exists d_2 \in \cS_2 \st d_1 \entails d_2. \]
\form#300:$\{\true\}$
\form#301:$ \nonredmap \bigl( \{\, d_1 \meet d_2 \mid d_1 \in \cS_1, d_2 \in \cS_2 \,\} \bigr) $
\form#302:$\nonredmap(\cS_1 \union \cS_2)$
\form#303:$\cS \in D_{\smallP}$
\form#304:$d \in D$
\form#305:$\nonredmap\bigl(\cS \union \{d\}\bigr)$
\form#306:$(\Pset_n)_{\smallP}$
\form#307:$\cS_1, \cS_2 \in (\Pset_n)_{\smallP}$
\form#308:$\cS_2$
\form#309:$\cS \in (\Pset_n)_{\smallP}$
\form#310:$\cP, \cQ$
\form#311:$\cP \uplus \cQ \neq \cP \union \cQ$
\form#312:$a, b, f \in \Rset$
\form#313:$a \equiv_f b$
\form#314:$\exists \mu \in \Zset \st a - b = \mu f$
\form#315:$\Sset \in \{ \Qset, \Rset \}$
\form#316:$\vect{a} \in \Sset^n \setdiff \{\vect{0}\}$
\form#317:$b, f \in \Sset$
\form#318:$\langle \vect{a}, \vect{x} \rangle \equiv_f b$
\form#319:$\Sset^n$
\form#320:\[ \bigl\{\, \vect{v} \in \Rset^n \bigm| \exists \mu \in \Zset \st \langle \vect{a}, \vect{v} \rangle = b + \mu f \,\bigr\}; \]
\form#321:$f \neq 0$
\form#322:$\langle \vect{a}, \vect{x} \rangle \equiv_0 b$
\form#323:$f = 0$
\form#324:$f$
\form#325:\[ \big\{\, \bigl(\langle \vect{a}, \vect{x} \rangle = b + \mu f\bigr) \bigm| \mu \in \Zset \,\bigr\}; \]
\form#326:$b \equiv_f 0$
\form#327:$\langle \vect{0}, \vect{x} \rangle \equiv_f b$
\form#328:$\cL \sseq \Rset^n$
\form#329:$\cL$
\form#330:$\Qset^n$
\form#331:$\cL = \emptyset$
\form#332:$\Gset_{n}$
\form#333:$\bigl\{\langle\vect{0}, \vect{x}\rangle \equiv_0 1\bigr\}$
\form#334:$0 = 1$
\form#335:$\bigl\{\langle\vect{a}, \vect{x}\rangle \equiv_2 0, \langle\vect{a}, \vect{x}\rangle \equiv_2 1\bigr\}$
\form#336:$\Gset_n$
\form#337:$\mu_1, \ldots, \mu_k \in \Zset$
\form#338:$\vect{v} = \sum_{j=1}^k \mu_j \vect{x}_j$
\form#339:$\inthull(S)$
\form#340:$\intaffinehull(S)$
\form#341:$\vect{p} \in \cL$
\form#342:$\vect{q} \in \Rset^n$
\form#343:$\vect{q} \neq \vect{0}$
\form#344:$\cL \neq \emptyset$
\form#345:$\vect{p} + \mu \vect{q} \in \cL$
\form#346:$\mu \in \Zset$
\form#347:$\vect{p} + \lambda \vect{l} \in \cL$
\form#348:$\lambda \in \Rset$
\form#349:$L, Q, P$
\form#350:\[ \cL = \linearhull(L) + \inthull(Q) + \intaffinehull(P) \]
\form#351:$\cL \in \Gset_n$
\form#352:$(L, Q, P)$
\form#353:$\cL = \ggen(L, Q, P)$
\form#354:$\cL = \ggen(L, Q, P) = \emptyset$
\form#355:$P \neq \emptyset$
\form#356:$\cL = \ggen(L, \emptyset, Q_{\vect{p}} \union P)$
\form#357:$\vect{p} \in P$
\form#358:$Q_{\vect{p}} = \{\, \vect{p} + \vect{q} \mid \vect{q} \in Q \,\}$
\form#359:$\cC'$
\form#360:$\card \cC \leq \card \cC'$
\form#361:$\cG = (L, Q, P)$
\form#362:$\cG' = (L', Q', P')$
\form#363:$\card L \leq \card L'$
\form#364:$\card Q + \card P \leq \card Q' + \card P'$
\form#365:$\pdim(\cG) = k$
\form#366:$0 \leq \pdim(\cG) \leq n$
\form#367:$\cL_1, \cL_2 \in \Gset_n$
\form#368:$\cL_1$
\form#369:$\cL_2$
\form#370:$\cL_1 \inters \cL_2$
\form#371:$\cL_1 \uplus \cL_2$
\form#372:$\cL_1 \in \Gset_n$
\form#373:$\cL_2 \in \Gset_m$
\form#374:$\Gset_{n+m}$
\form#375:\[ \Bigl\{\, (x_0, \ldots, x_{n-1}, y_0, \ldots, y_{m-1})^\transpose \in \Rset^{n+m} \Bigm| (x_0, \ldots, x_{n-1})^\transpose \in \cL_1, (y_0, \ldots, y_{m-1})^\transpose \in \cL_2 \,\Bigl\}. \]
\form#376:$\Gset_{n+i}$
\form#377:$\cL \sseq \Rset^2$
\form#378:\[ \bigl\{\, (x_0, x_1, x_2)^\transpose \in \Rset^3 \bigm| (x_0, x_1)^\transpose \in \cL \,\bigr\}. \]
\form#379:\[ \bigl\{\, (x_0, x_1, 0)^\transpose \in \Rset^3 \bigm| (x_0, x_1)^\transpose \in \cL \,\bigr\}. \]
\form#380:$\Gset_m$
\form#381:\[ \rho\bigl(\{0, \ldots, n-1\}\bigr) = \{0, \ldots, m-1\} \]
\form#382:\[ \Bigl\{\, \bigl(v_{\rho^{-1}(0)}, \ldots, v_{\rho^{-1}(m-1)}\bigr)^\transpose \Bigm| (v_0, \ldots, v_{n-1})^\transpose \in \cL \,\Bigr\}. \]
\form#383:\[ \sset{ \vect{u} \in \Rset^{n+m} }{ \exists \vect{v}, \vect{w} \in \cL \st u_i = v_i \\ \qquad \mathord{} \land \forall j = n, n+1, \ldots, n+m-1 \itc u_j = w_i \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_k = v_k = w_k }. \]
\form#384:$J$
\form#385:$\{0, \ldots, n-1\}$
\form#386:$m = \card J$
\form#387:\[ \cL_0 \uplus \cdots \uplus \cL_m \]
\form#388:\[ \cL_m \defeq \sset{ \vect{u} \in \Rset^{n-m} }{ \exists \vect{v} \in \cL \st u_{i'} = v_i \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_{k'} = v_k }, \]
\form#389:\[ \cL_d \defeq \sset{ \vect{u} \in \Rset^{n-m} }{ \exists \vect{v} \in \cL \st u_{i'} = v_{j_d} \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_{k'} = v_k } \]
\form#390:\[ k' \defeq k - \card \{\, j \in J \mid k > j \,\}. \]
\form#391:$x_k$
\form#392:$\mathrm{expr} = \langle \vect{a}, \vect{x} \rangle + b$
\form#393:$\bigl\{ (0, 0)^\transpose, (0, 3)^\transpose, (3, 0)^\transpose \bigr\}$
\form#394:$3x_0 + 2 x_1 + 1$
\form#395:$a_0 = 3, a_1 = 2, b = 1$
\form#396:$\bigl\{ (1, 0)^\transpose, (7, 3)^\transpose, (10, 0)^\transpose \bigr\}$
\form#397:$(1, 0)$
\form#398:$(3, -3), (0, 9)$
\form#399:$\{x \equiv_3 1, x + y \equiv_9 1\}$
\form#400:$x = y$
\form#401:$y$
\form#402:$\fund{\phi = (\mathrm{lhs}', \mathrm{rhs}, f)}{\Rset^n}{\Rset^n}$
\form#403:$f \in \Qset$
\form#404:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n \itc (\vect{v}, \vect{w}) \in \phi \iff \bigl( \langle \vect{c}, \vect{w} \rangle + d \equiv_f \langle \vect{a}, \vect{v} \rangle + b \bigr) \land \Bigl( \bigland_{0 \leq i < n, c_i = 0} w_i = v_i \Bigr). \]
\form#405:$ \cL_1 \nearrow \cL_2$
\form#406:\[ \bigl\{\, \vect{p} + \mu \vect{q} \in \Rset^n \bigm| \vect{p} \in \cL_1, \vect{q} \in \cL_2, \mu \in \Zset \,\bigr\}. \]
\form#407:$ \cg = \bigl( \langle \vect{a}, \vect{x} \rangle \equiv_f b \bigr) $
\form#408:$\cL_{\cg} = \gcon\bigl(\{\cg\}\bigr)$
\form#409:$\cg$
\form#410:$\cL \inters \cL_{\cg} = \emptyset$
\form#411:$\cL \inters \cL_{\cg} \neq \emptyset$
\form#412:$\cL \inters \cL_{\cg} \subset \cL$
\form#413:$\cL \sseq \cL_{\cg}$
\form#414:$\Qset$
\form#415:$\bigl(\langle \vect{e}_i, \vect{x} \rangle \equiv_f b\bigr)$
\form#416:$\bigl(\langle \vect{0}, \vect{x} \rangle \equiv_0 1\bigr)$
\form#417:$(0 = 1)$
\form#418:$B$
\form#419:$\cL = \gcon(\cC)$
\form#420:$\cL_1 \widen \cL_2$
\form#421:$\cL_1 \sseq \cL_2$
\form#422:$1$
\form#423:$2$
\form#424:$i+1$
\form#425:$x$
\form#426:$(0, 1)^\transpose \in \Rset^2$
\form#427:\[ \bigl\{\, (x, 0)^\transpose \in \Rset^2 \bigm| x \geq 0 \,\bigr\}, \]
\form#428:\[ \bigl\{\, (x, 0)^\transpose \in \Rset^2 \bigm| x \pmod{1}\ 0 \,\bigr\}, \]
\form#429:$x + y = 2$
\form#430:$x + y = 1 \pmod{2}$
\form#431:$\Zset^2$
\form#432:\[ \sum_{i=0}^{n-1} a_i x_i + b \]
\form#433:$x_i$
\form#434:$4x - 2y - z + 14$
\form#435:$3$
\form#436:$\sum_{i=0}^{n-1} a_i x_i + b = 0$
\form#437:$\sum_{i=0}^{n-1} a_i x_i + b \geq 0$
\form#438:$\sum_{i=0}^{n-1} a_i x_i + b > 0$
\form#439:$3x + 5y - z = 0$
\form#440:$4x \geq 2y - 13$
\form#441:$4x > 2y - 13$
\form#442:$x - 5y + 3z \leq 4$
\form#443:$x - 5y + 3z > 4$
\form#444:$\vect{l} = (a_0, \ldots, a_{n-1})^\transpose$
\form#445:$\vect{r} = (a_0, \ldots, a_{n-1})^\transpose$
\form#446:$\vect{p} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#447:$\vect{c} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#448:$d > 0$
\form#449:$x-y-z$
\form#450:$\vect{p} = (1, 0, 2)^\transpose \in \Rset^3$
\form#451:$\vect{0} \in \Rset^3$
\form#452:$\vect{0} \in \Rset^2$
\form#453:$\vect{0} \in \Rset^0$
\form#454:$\vect{p}$
\form#455:$\vect{q} = (-1.5, 3.2, 2.1)^\transpose \in \Rset^3$
\form#456:$\vect{c} = (1, 0, 2)^\transpose \in \Rset^3$
\form#457:$(a_0, \ldots, a_{n-1})^\transpose$
\form#458:$(a_0, 2 a_1, \ldots, (i+1)a_i, \ldots, n a_{n-1})^\transpose$
\form#459:$\vect{q} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#460:$\vect{p} = (1, -1, -1)^\transpose \in \Rset^3$
\form#461:$\vect{p1} = (-1.5, 3.2, 2.1)^\transpose \in \Rset^3$
\form#462:$\vect{q} = (1, 0, 2)^\transpose \in \Rset^3$
\form#463:$\cg = \sum_{i=0}^{n-1} a_i x_i + b = 0 \pmod m$
\form#464:$m \neq 0$
\form#465:$\pmod{1}$
\form#466:$4x = 2y - 13 \pmod{1}$
\form#467:$4x = 2y - 13 \pmod{2}$
\form#468:$x - 5y + 3z = 4 \pmod{5}$
\form#469:$2 e$
\form#470:$2x - 10y + 6z = 8 \pmod{5}$
\form#471:$R^0$
\form#472:$\{ 2 \} \sseq \Rset$
\form#473:\[ \bigl\{\, (2, y)^\transpose \in \Rset^2 \bigm| y \in \Rset \,\bigr\}. \]
\form#474:$\bigl\{ (2, 0)^\transpose \bigr\} \sseq \Rset^2$
\form#475:$x+4$
\form#476:$x+y$
\form#477:$x-y$
\form#478:$\bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2$
\form#479:$\bigl\{(3, 0)^\transpose \bigr\} \sseq \Rset^2$
\form#480:$w$
\form#481:$x+3$
\form#482:$(x, y)$
\form#483:$x=y$
\form#484:$x=-y$
\form#485:$\pm x_i \leq k$
\form#486:$x_i - x_j \leq k$
\form#487:$+\infty$
\form#488:\[ a_i x_i - a_j x_j \relsym b \]
\form#489:$\mathord{\relsym} \in \{ \leq, =, \geq \}$
\form#490:$a_j$
\form#491:$a_i = 0$
\form#492:$a_j = 0$
\form#493:$a_i = a_j$
\form#494:$3x - 3y \leq 1$
\form#495:$x - y \leq 1$
\form#496:$x - y \leq \frac{1}{3}$
\form#497:$x - y \leq k$
\form#498:$k > \frac{1}{3}$
\form#499:$\frac{1}{3}$
\form#500:$3x - y \leq 1$
\form#501:$\Rset^3$
\form#502:$-\infty$
\form#503:$0$
\form#504:$15 \cdot 10^2 = 1500$
\form#505:$5$
\form#506:$3/580000$
\form#507:$713/10$
\form#508:$-1929/15625$
\form#509:$11/50$
\form#510:$-220001/100$
\form#511:$35$
\form#512:$44027$
\form#513:$7/2$
\form#514:$2000$
\form#515:$1024$
\form#516:$1088$
\form#517:$9073863231288$
\form#518:$17/8$
\form#519:$482$
\form#520:$256$
\form#521:$(-\infty, +\infty) = \Qset$
\form#522:$r \times c$
\form#523:$M$
\form#524:$(r+n) \times c$
\form#525:$M \choose 0$
\form#526:$r \times (c+n)$
\form#527:$(M \, 0)$
\form#528:$(r+n) \times (c+m)$
\form#529:$\bigl({M \atop 0}{0 \atop 0}\bigr)$
\form#530:$(r+1) \times c$
\form#531:$\bigl({M \atop 0}\bigr)$
\form#532:$a_n$
\form#533:$\times$
\form#534:$\epsilon$
\form#535:$M \in \Rset^r \times \Rset^c$
\form#536:$N \in \Rset^{r+n} \times \Rset^{c+n}$
\form#537:$N = \bigl(\genfrac{}{}{0pt}{}{0}{M}\genfrac{}{}{0pt}{}{J}{o}\bigr)$
\form#538:$n \times n$
\form#539:$\sum_{i = 0}^{n - 1} a_i x_i + b$
\form#540:${a'}_{ij}$
\form#541:$a_{ij}$
\form#542:\[ {a'}_{ij} = \begin{cases} a_{ij} * \mathrm{denominator} + a_{iv} * \mathrm{expr}[j] \quad \text{for } j \neq v; \\ \mathrm{expr}[v] * a_{iv} \quad \text{for } j = v. \end{cases} \]
\form#543:$I$
\form#544:$d$
\form#545:$n/d$
\form#546:$0/1$
\form#547:$[n/d, +\infty)$
\form#548:$(n/d, +\infty)$
\form#549:$(-\infty, n/d]$
\form#550:$(-\infty, n/d)$
\form#551:\[ \frac{\sum_{i = 0}^{n - 1} a_i x_i + b} {\mathrm{denominator}}. \]
\form#552:$(r+dims) \times (c+dims)$
\form#553:$\bigl({A \atop 0}{0 \atop B}\bigr)$
\form#554:$dims \times dims$
\form#555:$\bigl({1 \atop 0}{0 \atop 1}\bigr)$
\form#556:$ e_1 = e_2 $
\form#557:$ e_1 = e_2 \pmod{0}$
\form#558:$\bigl({0 \atop A}{B \atop A}\bigr)$
\form#559:$\bigl({0 \atop 1}{1 \atop 0}\bigr)$
\form#560:$c = \bigl(\sum_{i=0}^{n-1} a_i x_i + b \relsym 0\bigr)$
\form#561:$\sum_{i=0}^{n-1} a_i x_i + b$
\form#562:$g = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#563:$d = 1$
\form#564:$\sum_{i=0}^{n-1} a_i x_i$
\form#565:$cg = \bigl(\sum_{i=0}^{n-1} a_i x_i + b = 0 \pmod{m}\bigr)$
\form#566:$0 \leq 1$
\form#567:$\sum_{i=0}^{n-1} 0 x_i + 0 = 0$
\form#568:$\sum_{i=0}^{n-1} 0 x_i + b \geq 0$
\form#569:$b \geq 0$
\form#570:$\sum_{i=0}^{n-1} 0 x_i + b > 0$
\form#571:$b > 0$
\form#572:$\sum_{i=0}^{n-1} 0 x_i + b = 0$
\form#573:$b \neq 0$
\form#574:$b < 0$
\form#575:$b \leq 0$
\form#576:$\epsilon \geq 0$
\form#577:$\epsilon \leq 1$
\form#578:$e1 = e2 \pmod{1}$
\form#579:$e = n \pmod{1}$
\form#580:$ e_1 = e_2 \pmod{m}$
\form#581:$ e_1 = e_2 \pmod{mk}$
\form#582:$\sum_{i=0}^{n-1} 0 x_i + 0 == 0$
\form#583:$\sum_{i=0}^{n-1} 0 x_i + b \%= 0 / m$
\form#584:$\sum_{i=0}^{n-1} 0 x_i + b == 0$
\form#585:$b \neq 0 \pmod{m}$
\form#586:$0 = 1 \pmod{1}$
\form#587:$0 = 1 \pmod{0}$
\form#588:$\frac{num}{den}$
\form#589:$\frac{d_{j}}{\|\Delta x^{j} \|}$
\form#590:\[ \|\Delta x^{j} \| = \left( 1+\sum_{i=1}^{m} \alpha_{ij}^2 \right)^{\frac{1}{2}}. \]
\form#591:$\alpha$
\form#592:$\mathrm{var}$
\form#593:$u$
\form#594:$l$
\form#595:$[l, +\mathrm{infty})$
\form#596:$\mathrm{var} == l$
\form#597:$(l, +\mathrm{infty})$
\form#598:$(-\mathrm{infty}, u]$
\form#599:$\mathrm{var} == u$
\form#600:$(-\mathrm{infty}, u)$
\form#601:$\mathrm{var}' \relsym \frac{\mathrm{expr}}{\mathrm{denominator}}$
\form#602:$\mathrm{lhs}' \relsym \mathrm{rhs}$
\form#603:$\frac{\mathrm{lb\_expr}}{\mathrm{denominator}} \leq \mathrm{var}' \leq \frac{\mathrm{ub\_expr}}{\mathrm{denominator}}$
\form#604:$\pm x \leq r$
\form#605:$\pm x < r$
\form#606:$r \in \Qset$
\form#607:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cP \,\bigr\}. \]
\form#608:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cP \,\bigr\}. \]
\form#609:$f(k)$
\form#610:$\dots$
\form#611:$\Rset^0 = \{\cdot\}$
\form#612:\[ \begin{cases} sat\_c[i][j] = 0, \quad \text{if } G[i] \cdot C^\mathrm{T}[j] = 0; \\ sat\_c[i][j] = 1, \quad \text{if } G[i] \cdot C^\mathrm{T}[j] > 0. \end{cases} \]
\form#613:\[ \begin{cases} sat\_g[i][j] = 0, \quad \text{if } C[i] \cdot G^\mathrm{T}[j] = 0; \\ sat\_g[i][j] = 1, \quad \text{if } C[i] \cdot G^\mathrm{T}[j] > 0. \end{cases} \]
\form#614:$\mathrm{var}' = \frac{\mathrm{expr}}{\mathrm{denominator}} \pmod{\mathrm{modulus}}$
\form#615:$\mathrm{lhs}' = \mathrm{rhs} \pmod{\mathrm{modulus}}$
\form#616:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cL \,\bigr\}. \]
\form#617:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cL \,\bigr\}. \]
\form#618:$L_\infty$
\form#619:$\mathtt{y} \Delta \mathtt{x}$
\form#620:$\cB \sseq \Rset^2$
\form#621:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cB \,\bigr\}. \]
\form#622:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cB \,\bigr\}. \]
\form#623:$B \sseq \Rset^n$
\form#624:$D \sseq \Rset^m$
\form#625:$R \sseq \Rset^{n+m}$
\form#626:\[ R \defeq \Bigl\{\, (x_1, \ldots, x_n, y_1, \ldots, y_m)^\transpose \Bigm| (x_1, \ldots, x_n)^\transpose \in B, (y_1, \ldots, y_m)^\transpose \in D \,\Bigl\}. \]
\form#627:$\frac{\mathrm{lb}}{\mathrm{d}} \leq \mathrm{var}' \leq \frac{\mathrm{ub}}{\mathrm{d}}$
\form#628:$\mathrm{var}' \relsym \frac{\mathrm{le}}{\mathrm{d}}$
\form#629:$\pm v \leq r$
\form#630:$\pm v < r$
\form#631:$e < 0$
\form#632:$e \leq 0$
\form#633:$e = 0$
\form#634:$e \geq 0$
\form#635:$e > 0$
\form#636:$k-1$
\form#637:$4x + y - 2z >= 5$
\form#638:$(-12.5, -0.625, 0)^\transpose$
\form#639:$ (\cP \nearrow \cQ)$
\form#640:$t_1$
\form#641:$t_2$
\form#642:$t_1 > 0 $
\form#643:$t_2 = t_1 - 1$
\form#644:$t_2 = t_1$
\form#645:$t_1 = 0$
\form#646:$t_2 = 0$
\form#647:$\mathrm{LP}$
