\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:$\cP_1, \cP_2 \in \Pset_n$
\form#111:$\cP_1$
\form#112:$\cP_2$
\form#113:$\cP_1 \inters \cP_2$
\form#114:$\cP_1 \uplus \cP_2$
\form#115:$\cP_1, \cP_2 \in \CPset_n$
\form#116:$\cQ \in \Pset_m$
\form#117:$\cR \in \Pset_{n+m}$
\form#118:\[ \cR \defeq \Bigl\{\, (x_0, \ldots, x_{n-1}, y_0, \ldots, y_{m-1})^\transpose \in \Rset^{n+m} \Bigm| (x_0, \ldots, x_{n-1})^\transpose \in \cP, (y_0, \ldots, y_{m-1})^\transpose \in \cQ \,\Bigl\}. \]
\form#119:$n+m$
\form#120:$\cQ$
\form#121:$\cQ \in \Pset_{n+i}$
\form#122:$i+n$
\form#123:$\cP \sseq \Rset^2$
\form#124:\[ \cQ = \bigl\{\, (x_0, x_1, x_2)^\transpose \in \Rset^3 \bigm| (x_0, x_1)^\transpose \in \cP \,\bigr\}. \]
\form#125:\[ \cQ = \bigl\{\, (x_0, x_1, 0)^\transpose \in \Rset^3 \bigm| (x_0, x_1)^\transpose \in \cP \,\bigr\}. \]
\form#126:$\cP \in \Pset_4$
\form#127:$\bigl\{ (3, 1, 0, 2)^\transpose \bigr\} \sseq \Rset^4$
\form#128:$\{x_1, x_2\}$
\form#129:\[ \cQ = \bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2. \]
\form#130:$m$
\form#131:$m = 2$
\form#132:\[ \cQ = \bigl\{ (3, 1)^\transpose \bigr\} \sseq \Rset^2. \]
\form#133:$\pard{\rho}{\{0, \ldots, n-1\}}{\Nset}$
\form#134:$\rho\bigl(\{0, \ldots, n-1\}\bigr) = \{0, \ldots, m-1\}$
\form#135:$\rho$
\form#136:$m = 0$
\form#137:$\Rset^0$
\form#138:\[ \cQ \defeq \Bigl\{\, \bigl(v_{\rho^{-1}(0)}, \ldots, v_{\rho^{-1}(m-1)}\bigr)^\transpose \Bigm| (v_0, \ldots, v_{n-1})^\transpose \in \cP \,\Bigr\}. \]
\form#139:$n > 0$
\form#140:$n+1$
\form#141:$\ldots$
\form#142:$n+m-1$
\form#143:\[ \cQ \defeq \sset{ \vect{u} \in \Rset^{n+m} }{ \exists \vect{v}, \vect{w} \in \cP \st u_i = v_i \\ \qquad \mathord{} \land \forall j = n, n+1, \ldots, n+m-1 \itc u_j = w_i \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_k = v_k = w_k }. \]
\form#144:$J = \{ j_0, \ldots, j_{m-1} \}$
\form#145:$m < n$
\form#146:$j < n$
\form#147:$j \in J$
\form#148:$i < n$
\form#149:$i \notin J$
\form#150:\[ \cQ \defeq \biguplus_{d = 0}^m \cQ_d \]
\form#151:\[ \cQ_m \defeq \sset{ \vect{u} \in \Rset^{n-m} }{ \exists \vect{v} \in \cP \st u_{i'} = v_i \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_{k'} = v_k } \]
\form#152:$ d = 0 $
\form#153:$ \ldots $
\form#154:$ m-1 $
\form#155:\[ \cQ_d \defeq \sset{ \vect{u} \in \Rset^{n-m} }{ \exists \vect{v} \in \cP \st u_{i'} = v_{j_d} \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_{k'} = v_k }, \]
\form#156:$ k = 0 $
\form#157:$ n-1 $
\form#158:\[ k' \defeq k - \card \{\, j \in J \mid k > j \,\}, \]
\form#159:$\card S$
\form#160:$\fund{\phi}{\Rset^n}{\Rset^m}$
\form#161:$\phi(S) \sseq \Rset^m$
\form#162:$\phi$
\form#163:$S \sseq \Rset^n$
\form#164:\[ \phi(S) \defeq \bigl\{\, \phi(\vect{v}) \in \Rset^m \bigm| \vect{v} \in S \,\bigr\}. \]
\form#165:$\phi^{-1}(S') \sseq \Rset^n$
\form#166:$S' \sseq \Rset^m$
\form#167:$\phi(S) \sseq S'$
\form#168:\[ \phi^{-1}(S') \defeq \bigl\{\, \vect{v} \in \Rset^n \bigm| \phi(\vect{v}) \in S' \,\bigr\}. \]
\form#169:$A \in \Rset^m \times \Rset^n$
\form#170:$\vect{b} \in \Rset^m$
\form#171:$\vect{x} \in \Rset^n$
\form#172:$\phi(\vect{x}) = A\vect{x} + \vect{b}$
\form#173:$n = m$
\form#174:$x_k$
\form#175:$\mathrm{expr} = \sum_{i=0}^{n-1} a_i x_i + b$
\form#176:\[ A = \begin{pmatrix} 1 & & 0 & 0 & \cdots & \cdots & 0 \\ & \ddots & & \vdots & & & \vdots \\ 0 & & 1 & 0 & \cdots & \cdots & 0 \\ a_0 & \cdots & a_{k-1} & a_k & a_{k+1} & \cdots & a_{n-1} \\ 0 & \cdots & \cdots & 0 & 1 & & 0 \\ \vdots & & & \vdots & & \ddots & \\ 0 & \cdots & \cdots & 0 & 0 & & 1 \end{pmatrix}, \qquad \vect{b} = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ b \\ 0 \\ \vdots \\ 0 \end{pmatrix} \]
\form#177:$a_i$
\form#178:$b$
\form#179:$(k+1)$
\form#180:$A$
\form#181:$\vect{b}$
\form#182:$(x_0, \ldots, x_{n-1})^\transpose$
\form#183:\[ \Bigl(x_0, \ldots, \bigl(\textstyle{\sum_{i=0}^{n-1}} a_i x_i + b\bigr), \ldots, x_{n-1}\Bigr)^\transpose. \]
\form#184:$\bigl\{ (0, 0)^\transpose, (0, 3)^\transpose, (3, 0)^\transpose, (3, 3)^\transpose \bigr\}$
\form#185:$x_0$
\form#186:$x_0 + 2 x_1 + 4$
\form#187:$k = 0$
\form#188:$a_0 = 1, a_1 = 2, b = 4$
\form#189:$\bigl\{ (4, 0)^\transpose, (10, 3)^\transpose, (7, 0)^\transpose, (13, 3)^\transpose \bigr\}$
\form#190:$x_0 - 2 x_1$
\form#191:$x_1$
\form#192:$a_0 = 0, a_1 = 1, b = 0$
\form#193:$a_k$
\form#194:$\mathrm{lhs} = \sum_{i=0}^{n-1} a'_i x_i + b'$
\form#195:$\mathord{\relsym} \in \{ <, \leq, =, \geq, > \}$
\form#196:$\mathrm{rhs} = \sum_{i=0}^{n-1} a_i x_i + b$
\form#197:$\mathrm{lhs} \relsym \mathrm{rhs}$
\form#198:\[ \sset{ (w_0, \ldots, w_{n-1})^\transpose \in \Rset^n }{ (v_0, \ldots, v_{n-1})^\transpose \in \cP, \\ \bigl( i \in \{0, \ldots, n-1\} \land a'_i = 0 \implies w_i = v_i \bigr), \\ \sum_{i=0}^{n-1} a'_i w_i + b' \relsym \sum_{i=0}^{n-1} a_i v_i + b }. \]
\form#199:$\mathrm{lhs} = x_k$
\form#200:$\mathrm{rhs}$
\form#201:$\cP, \cQ \in \Pset_n$
\form#202:$ \cP \nearrow \cQ$
\form#203:\[ \bigl\{\, \vect{p} + \lambda \vect{q} \in \Rset^n \bigm| \vect{p} \in \cP, \vect{q} \in \cQ, \lambda \in \nonnegRset \,\bigr\}. \]
\form#204:$\cP,\cQ \in \CPset_n$
\form#205:$ c = \bigl( \langle \vect{a}, \vect{x} \rangle \relsym b \bigr) $
\form#206:$c$
\form#207:$\cP \inters \cQ = \emptyset$
\form#208:$\cP \inters \cQ \neq \emptyset$
\form#209:$\cP \inters \cQ \subset \cP$
\form#210:$\cP \sseq \cQ$
\form#211:$\cP \sseq \cH$
\form#212:$\cH$
\form#213:$g$
\form#214:$\cB$
\form#215:$\vect{e}_i = (0, \ldots, 1, \ldots, 0)^\transpose$
\form#216:$\langle \vect{e}_i, \vect{x} \rangle \relsym b$
\form#217:$\mathord{\relsym}$
\form#218:$\mathord{\geq}$
\form#219:$\mathord{>}$
\form#220:$\langle\vect{e}_i,\vect{x}\rangle \relsym b$
\form#221:$\mathord{\leq}$
\form#222:$\mathord{<}$
\form#223:$\cQ \in \Pset_n$
\form#224:$\cQ \sseq \cP$
\form#225:$k$
\form#226:$D$
\form#227:$\mathord{\entails}$
\form#228:$\mathord{\meet}$
\form#229:$\true$
\form#230:$\false$
\form#231:$\cS \in \wp(D)$
\form#232:$\false \notin \cS$
\form#233:$\forall d_1, d_2 \in \cS \itc d_1 \entails d_2 \implies d_1 = d_2$
\form#234:$\wpfn{D}{\entails}$
\form#235:$\fund{\nonredmap}{\wpf(D)}{\wpfn{D}{\entails}}$
\form#236:$\cS \in \wpf(D)$
\form#237:\[ \nonredmap(\cS) \defeq \cS \setdiff \{\, d \in \cS \mid d = \false \text{ or } \exists d' \in \cS \st d \sentails d' \,\}. \]
\form#238:$\cS \neq \{ \false \}$
\form#239:$\nonredmap(\cS)$
\form#240:$\cS$
\form#241:$D_{\smallP}$
\form#242:$\mathord{\entailsP}$
\form#243:$\cS_1 \entailsP \cS_2$
\form#244:\[ \forall d_1 \in \cS_1 \itc \exists d_2 \in \cS_2 \st d_1 \entails d_2. \]
\form#245:$\{\true\}$
\form#246:$\cS_1$
\form#247:$\cS_2 \in D_{\smallP}$
\form#248:$\nonredmap\bigl(\{d_1 \meet d_2 \,|\, d_1 \in \cS_1, d_2 \in \cS_2\}\bigr)$
\form#249:$\nonredmap(\cS_1 \union \cS_2)$
\form#250:$\cS \in D_{\smallP}$
\form#251:$d \in D$
\form#252:$\nonredmap\bigl(\cS \union \{d\}\bigr)$
\form#253:$(\Pset_n)_{\smallP}$
\form#254:$\cS_1, \cS_2 \in (\Pset_n)_{\smallP}$
\form#255:$\cS_2$
\form#256:$\cS \in (\Pset_n)_{\smallP}$
\form#257:$\cP, \cQ$
\form#258:$\cP \uplus \cQ \neq \cP \union \cQ$
\form#259:\[ \sum_{i=0}^{n-1} a_i x_i + b \]
\form#260:$x_i$
\form#261:$4x - 2y - z + 14$
\form#262:$3$
\form#263:$\sum_{i=0}^{n-1} a_i x_i + b = 0$
\form#264:$\sum_{i=0}^{n-1} a_i x_i + b \geq 0$
\form#265:$\sum_{i=0}^{n-1} a_i x_i + b > 0$
\form#266:$3x + 5y - z = 0$
\form#267:$4x \geq 2y - 13$
\form#268:$2$
\form#269:$4x > 2y - 13$
\form#270:$x - 5y + 3z <= 4$
\form#271:$x - 5y + 3z > 4$
\form#272:$\vect{l} = (a_0, \ldots, a_{n-1})^\transpose$
\form#273:$\vect{r} = (a_0, \ldots, a_{n-1})^\transpose$
\form#274:$\vect{p} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#275:$\vect{c} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#276:$d > 0$
\form#277:$x-y-z$
\form#278:$\vect{p} = (1, 0, 2)^\transpose \in \Rset^3$
\form#279:$\vect{0} \in \Rset^3$
\form#280:$\vect{0} \in \Rset^2$
\form#281:$\vect{0} \in \Rset^0$
\form#282:$\vect{p}$
\form#283:$\vect{q} = (-1.5, 3.2, 2.1)^\transpose \in \Rset^3$
\form#284:$\vect{c} = (1, 0, 2)^\transpose \in \Rset^3$
\form#285:$(a_0, \ldots, a_{n-1})^\transpose$
\form#286:$(a_0, 2 a_1, \ldots, (i+1)a_i, \ldots, n a_{n-1})^\transpose$
\form#287:$R^0$
\form#288:$\{ 2 \} \sseq \Rset$
\form#289:\[ \bigl\{\, (2, y)^\transpose \in \Rset^2 \bigm| y \in \Rset \,\bigr\}. \]
\form#290:$\bigl\{ (2, 0)^\transpose \bigr\} \sseq \Rset^2$
\form#291:$x$
\form#292:$x+4$
\form#293:$x+y$
\form#294:$x-y$
\form#295:$y$
\form#296:$\bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2$
\form#297:$\bigl\{(3, 0)^\transpose \bigr\} \sseq \Rset^2$
\form#298:$w$
\form#299:$0 = 1$
\form#300:$0 \leq 1$
\form#301:$I$
\form#302:$d$
\form#303:$n/d$
\form#304:$0/1$
\form#305:$[n/d, +\infty)$
\form#306:$(n/d, +\infty)$
\form#307:$(-\infty, n/d]$
\form#308:$(-\infty, n/d)$
\form#309:$\mathrm{var}' \relsym \frac{\mathrm{expr}}{\mathrm{denominator}}$
\form#310:$\mathrm{lhs}' \relsym \mathrm{rhs}$
\form#311:$\pm v \leq r$
\form#312:$\pm v < r$
\form#313:$r \in \Qset$
\form#314:$k-1$
\form#315:$\Rset^3$
\form#316:$4x + y - 2z >= 5$
\form#317:$(-12.5, -0.625, 0)^\transpose$
\form#318:$ (\cP \nearrow \cQ)$
\form#319:$\pm x \leq r$
\form#320:$\pm x < r$
\form#321:$c = \bigl(\sum_{i=0}^{n-1} a_i x_i + b \relsym 0\bigr)$
\form#322:$\sum_{i=0}^{n-1} a_i x_i + b$
\form#323:$g = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$
\form#324:$d = 1$
\form#325:$\sum_{i=0}^{n-1} a_i x_i$
\form#326:$\epsilon \geq 0$
\form#327:$\epsilon \leq 1$
\form#328:$\sum_{i=0}^{n-1} 0 x_i + 0 = 0$
\form#329:$\sum_{i=0}^{n-1} 0 x_i + b \geq 0$
\form#330:$b \geq 0$
\form#331:$\sum_{i=0}^{n-1} 0 x_i + b > 0$
\form#332:$b > 0$
\form#333:$\sum_{i=0}^{n-1} 0 x_i + b = 0$
\form#334:$b \neq 0$
\form#335:$b < 0$
\form#336:$b \leq 0$
\form#337:$-1$
\form#338:$0$
\form#339:$1$
\form#340:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cP \,\bigr\}. \]
\form#341:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cP \,\bigr\}. \]
\form#342:$f$
\form#343:$f(k)$
\form#344:$\dots$
\form#345:$\Rset^0 = \{\cdot\}$
\form#346:$\epsilon$
\form#347:\[ \begin{cases} sat\_c[i][j] = 0, \quad \text{if } G[i] \cdot C^\mathrm{T}[j] = 0; \\ sat\_c[i][j] = 1, \quad \text{if } G[i] \cdot C^\mathrm{T}[j] > 0. \end{cases} \]
\form#348:\[ \begin{cases} sat\_g[i][j] = 0, \quad \text{if } C[i] \cdot G^\mathrm{T}[j] = 0; \\ sat\_g[i][j] = 1, \quad \text{if } C[i] \cdot G^\mathrm{T}[j] > 0. \end{cases} \]
\form#349:$e < 0$
\form#350:$e \leq 0$
\form#351:$e = 0$
\form#352:$e \geq 0$
\form#353:$e > 0$
