\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:$\phi = (\mathrm{lhs}' \relsym \mathrm{rhs})$ \form#258:$\mathrm{lhs} = \langle \vect{c}, \vect{x} \rangle + d$ \form#259:$\mathrm{rhs} = \langle \vect{a}, \vect{x} \rangle + b$ \form#260:$\mathord{\relsym} \in \{ <, \leq, =, \geq, > \}$ \form#261:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n \itc (\vect{v}, \vect{w}) \in \phi \iff \bigl( \langle \vect{c}, \vect{w} \rangle + d \relsym \langle \vect{a}, \vect{v} \rangle + b \bigr) \land \Bigl( \bigland_{0 \leq i < n, c_i = 0} w_i = v_i \Bigr). \] \form#262:$\mathrm{lhs} = x_k$ \form#263:$x'_k = \mathrm{rhs}$ \form#264:$\mathrm{lhs}$ \form#265:$\mathrm{rhs}$ \form#266:$\mathrm{lhs}' \leq \mathrm{rhs}$ \form#267:$\mathrm{rhs}' \geq \mathrm{lhs}$ \form#268:$\cQ \in \Pset_n$ \form#269:$i \in \{ 0, \ldots, n-1 \}$ \form#270:\[ \cQ = \bigl\{\, \vect{w} \in \Rset^n \bigm| \exists \vect{v} \in \cP \st \forall j \in \{0, \ldots, n-1\} \itc j \neq i \implies w_j = v_j \,\bigr\}. \] \form#271:$\cP, \cQ \in \Pset_n$ \form#272:$ \cP \nearrow \cQ$ \form#273:\[ \bigl\{\, \vect{p} + \lambda \vect{q} \in \Rset^n \bigm| \vect{p} \in \cP, \vect{q} \in \cQ, \lambda \in \nonnegRset \,\bigr\}. \] \form#274:$\cP,\cQ \in \CPset_n$ \form#275:$\cP, \cQ, \cR \in \Pset_n$ \form#276:$\cR$ \form#277:$\cR \inters \cQ = \cP \inters \cQ$ \form#278:$\cR \Sseq \cP$ \form#279:$r \leq p$ \form#280:$r$ \form#281:$p$ \form#282:$ c = \bigl( \langle \vect{a}, \vect{x} \rangle \relsym b \bigr) $ \form#283:$c$ \form#284:$\cP \inters \cQ = \emptyset$ \form#285:$\cP \inters \cQ \neq \emptyset$ \form#286:$\cP \inters \cQ \subset \cP$ \form#287:$\cP \sseq \cQ$ \form#288:$\cP \sseq \cH$ \form#289:$\cH$ \form#290:$g$ \form#291:$\cP \widen \cQ$ \form#292:$\cP, \cQ \in \CPset_n$ \form#293:$\cP \Sseq \cP \widen \cQ$ \form#294:$\cQ \sseq \cP$ \form#295:$\cP = \cQ$ \form#296:$k$ \form#297:$\mathord{\relsym} \in \{ =, \geq, >\}$ \form#298:$k \in \{ 0, \ldots, i-1, i+1, \ldots, n-1 \}$ \form#299:$a_k = 0$ \form#300:$x = r$ \form#301:$x \leq r$ \form#302:$x \geq r$ \form#303:$x < r$ \form#304:$x > r$ \form#305:$r \in \Rset$ \form#306:$\cB$ \form#307:$\vect{e}_i = (0, \ldots, 1, \ldots, 0)^\transpose$ \form#308:$\langle \vect{e}_i, \vect{x} \rangle \relsym b$ \form#309:$\mathord{\relsym}$ \form#310:$\mathord{\geq}$ \form#311:$\mathord{>}$ \form#312:$\langle\vect{e}_i,\vect{x}\rangle \relsym b$ \form#313:$\mathord{\leq}$ \form#314:$\mathord{<}$ \form#315:$\{-2, -1, 0, 1, 2\}$ \form#316:$\mathord{\relsym} \in \{ =, \geq\}$ \form#317:$i, j \in \{ 0, \ldots, n-1 \}$ \form#318:$a_i, a_j \in \{ -1, 0, 1 \}$ \form#319:$a_i \neq a_j$ \form#320:$k \notin \{ i, j \}$ \form#321:$a, b, f \in \Rset$ \form#322:$a \equiv_f b$ \form#323:$\exists \mu \in \Zset \st a - b = \mu f$ \form#324:$\Sset \in \{ \Qset, \Rset \}$ \form#325:$\vect{a} \in \Sset^n \setdiff \{\vect{0}\}$ \form#326:$b, f \in \Sset$ \form#327:$\langle \vect{a}, \vect{x} \rangle \equiv_f b$ \form#328:$\Sset^n$ \form#329:\[ \bigl\{\, \vect{v} \in \Rset^n \bigm| \exists \mu \in \Zset \st \langle \vect{a}, \vect{v} \rangle = b + \mu f \,\bigr\}; \] \form#330:$f \neq 0$ \form#331:$\langle \vect{a}, \vect{x} \rangle \equiv_0 b$ \form#332:$f = 0$ \form#333:$f$ \form#334:\[ \big\{\, \bigl(\langle \vect{a}, \vect{x} \rangle = b + \mu f\bigr) \bigm| \mu \in \Zset \,\bigr\}; \] \form#335:$b \equiv_f 0$ \form#336:$\langle \vect{0}, \vect{x} \rangle \equiv_f b$ \form#337:$\cL \sseq \Rset^n$ \form#338:$\cL$ \form#339:$\Qset^n$ \form#340:$\cL = \emptyset$ \form#341:$\Gset_{n}$ \form#342:$\bigl\{\langle\vect{0}, \vect{x}\rangle \equiv_0 1\bigr\}$ \form#343:$0 = 1$ \form#344:$\bigl\{\langle\vect{a}, \vect{x}\rangle \equiv_2 0, \langle\vect{a}, \vect{x}\rangle \equiv_2 1\bigr\}$ \form#345:$\Gset_n$ \form#346:$\mu_1, \ldots, \mu_k \in \Zset$ \form#347:$\vect{v} = \sum_{j=1}^k \mu_j \vect{x}_j$ \form#348:$\inthull(S)$ \form#349:$\intaffinehull(S)$ \form#350:$\vect{p} \in \cL$ \form#351:$\vect{q} \in \Rset^n$ \form#352:$\vect{q} \neq \vect{0}$ \form#353:$\cL \neq \emptyset$ \form#354:$\vect{p} + \mu \vect{q} \in \cL$ \form#355:$\mu \in \Zset$ \form#356:$\vect{p} + \lambda \vect{l} \in \cL$ \form#357:$\lambda \in \Rset$ \form#358:$L, Q, P$ \form#359:\[ \cL = \linearhull(L) + \inthull(Q) + \intaffinehull(P) \] \form#360:$\cL \in \Gset_n$ \form#361:$(L, Q, P)$ \form#362:$\cL = \ggen(L, Q, P)$ \form#363:$\cL = \ggen(L, Q, P) = \emptyset$ \form#364:$P \neq \emptyset$ \form#365:$\cL = \ggen(L, \emptyset, Q_{\vect{p}} \union P)$ \form#366:$\vect{p} \in P$ \form#367:$Q_{\vect{p}} = \{\, \vect{p} + \vect{q} \mid \vect{q} \in Q \,\}$ \form#368:$\cC'$ \form#369:$\card \cC \leq \card \cC'$ \form#370:$\cG = (L, Q, P)$ \form#371:$\cG' = (L', Q', P')$ \form#372:$\card L \leq \card L'$ \form#373:$\card Q + \card P \leq \card Q' + \card P'$ \form#374:$\pdim(\cG) = k$ \form#375:$0 \leq \pdim(\cG) \leq n$ \form#376:$x_k$ \form#377:$\mathrm{expr} = \langle \vect{a}, \vect{x} \rangle + b$ \form#378:$\bigl\{ (0, 0)^\transpose, (0, 3)^\transpose, (3, 0)^\transpose \bigr\}$ \form#379:$3x_0 + 2 x_1 + 1$ \form#380:$a_0 = 3, a_1 = 2, b = 1$ \form#381:$\cL_1$ \form#382:$\bigl\{ (1, 0)^\transpose, (7, 3)^\transpose, (10, 0)^\transpose \bigr\}$ \form#383:$(1, 0)$ \form#384:$(3, -3), (0, 9)$ \form#385:$\{x \equiv_3 1, x + y \equiv_9 1\}$ \form#386:$\cL_2$ \form#387:$x = y$ \form#388:$y$ \form#389:$\fund{\phi = (\mathrm{lhs}', \mathrm{rhs}, f)}{\Rset^n}{\Rset^n}$ \form#390:$f \in \Qset$ \form#391:\[ \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n \itc (\vect{v}, \vect{w}) \in \phi \iff \bigl( \langle \vect{c}, \vect{w} \rangle + d \equiv_f \langle \vect{a}, \vect{v} \rangle + b \bigr) \land \Bigl( \bigland_{0 \leq i < n, c_i = 0} w_i = v_i \Bigr). \] \form#392:$\mathrm{expr} = \bigl(\langle \vect{a}, \vect{x} \rangle + b\bigr)$ \form#393:$c, f \in \Rset$ \form#394:$\cg = ( \mathrm{expr} \equiv_f c )$ \form#395:$\mathrm{expr}$ \form#396:$\mathrm{val} = \langle \vect{a}, \vect{w} \rangle + b$ \form#397:$\vect{w} \in \cL$ \form#398:\[ \lvert\mathrm{val}\rvert = \min\Bigl\{\, \bigl\lvert\langle \vect{a}, \vect{v} \rangle + b \bigr\rvert \Bigm| \vect{v} \in \cL \,\Bigr\}. \] \form#399:$( \mathrm{expr} = c )$ \form#400:$0$ \form#401:$\cL_1, \cL_2 \in \Gset_n$ \form#402:$ \cL_1 \nearrow \cL_2$ \form#403:\[ \bigl\{\, \vect{p} + \mu \vect{q} \in \Rset^n \bigm| \vect{p} \in \cL_1, \vect{q} \in \cL_2, \mu \in \Zset \,\bigr\}. \] \form#404:$ \cg = \bigl( \langle \vect{a}, \vect{x} \rangle \equiv_f b \bigr) $ \form#405:$\cL_{\cg} = \gcon\bigl(\{\cg\}\bigr)$ \form#406:$\cg$ \form#407:$\cL \inters \cL_{\cg} = \emptyset$ \form#408:$\cL \inters \cL_{\cg} \neq \emptyset$ \form#409:$\cL \inters \cL_{\cg} \subset \cL$ \form#410:$\cL \sseq \cL_{\cg}$ \form#411:$\cL \inters \cQ = \emptyset$ \form#412:$\cL \inters \cQ \neq \emptyset$ \form#413:$\cL \inters \cQ \subset \cL$ \form#414:$\cL \sseq \cQ$ \form#415:$=$ \form#416:$J$ \form#417:$\{0, \ldots, n-1\}$ \form#418:$w$ \form#419:$R = \{r \in \Rset \mid 0 \leq r < 2^w\}$ \form#420:$R = \{r \in \Rset \mid -2^{w-1} \leq r < 2^{w-1}\}$ \form#421:$v_j$ \form#422:$j$ \form#423:$\cL'$ \form#424:$a \in R$ \form#425:$a$ \form#426:$\cL' = \cL$ \form#427:$a + z \in R$ \form#428:$z \in \Zset$ \form#429:$(0, \ldots, 0, v_j, 0, \ldots, 0)$ \form#430:$v_j = 1$ \form#431:$v_j = a \mod 2^w$ \form#432:$a \in \Rset$ \form#433:$a'$ \form#434:$a' = a \mod 2^w$ \form#435:$a'\in R$ \form#436:$v_j = 2^w$ \form#437:$\cL_1 \widen \cL_2$ \form#438:$\cL_1 \sseq \cL_2$ \form#439:$D$ \form#440:$\mathord{\entails}$ \form#441:$\mathord{\meet}$ \form#442:$\true$ \form#443:$\false$ \form#444:$\cS \in \wp(D)$ \form#445:$\false \notin \cS$ \form#446:$\forall d_1, d_2 \in \cS \itc d_1 \entails d_2 \implies d_1 = d_2$ \form#447:$\wpfn{D}{\entails}$ \form#448:$\fund{\nonredmap}{\wpf(D)}{\wpfn{D}{\entails}}$ \form#449:$\cS \in \wpf(D)$ \form#450:\[ \nonredmap(\cS) \defeq \cS \setdiff \{\, d \in \cS \mid d = \false \text{ or } \exists d' \in \cS \st d \sentails d' \,\}. \] \form#451:$d \sentails d'$ \form#452:$d \entails d' \land d \ne d'$ \form#453:$\cS$ \form#454:$\nonredmap(\cS)$ \form#455:$\cS \neq \{ \false \}$ \form#456:$D_{\smallP}$ \form#457:$\mathord{\entailsP}$ \form#458:$\cS_1$ \form#459:$\cS_2 \in D_{\smallP}$ \form#460:$\cS_1 \entailsP \cS_2$ \form#461:\[ \forall d_1 \in \cS_1 \itc \exists d_2 \in \cS_2 \st d_1 \entails d_2. \] \form#462:$\{\true\}$ \form#463:$ \nonredmap \bigl( \{\, d_1 \meet d_2 \mid d_1 \in \cS_1, d_2 \in \cS_2 \,\} \bigr) $ \form#464:$\nonredmap(\cS_1 \union \cS_2)$ \form#465:$\cS \in D_{\smallP}$ \form#466:$d \in D$ \form#467:$\nonredmap\bigl(\cS \union \{d\}\bigr)$ \form#468:$\cS_1 = \{ d_1, \ldots, d_m \}$ \form#469:$\cS_2 = \{ c_1, \ldots, c_n \}$ \form#470:$\cS = \{ s_1, \ldots, s_q \}$ \form#471:$\cS_2$ \form#472:$q \leq m$ \form#473:$s_k \in \cS$ \form#474:$d_i \in \cS_1$ \form#475:$c_j \in \cS_2$ \form#476:$s_k$ \form#477:$d_i$ \form#478:$c_j$ \form#479:$\cS_1, \cS_2$ \form#480:$\uplus$ \form#481:$c, d$ \form#482:$c \uplus d \neq c \union d$ \form#483:$i+1$ \form#484:\[ \sum_{i=0}^{n-1} a_i x_i + b \] \form#485:$x_i$ \form#486:$4x - 2y - z + 14$ \form#487:$3$ \form#488:$1$ \form#489:$2$ \form#490:$\sum_{i=0}^{n-1} a_i x_i + b = 0$ \form#491:$\sum_{i=0}^{n-1} a_i x_i + b \geq 0$ \form#492:$\sum_{i=0}^{n-1} a_i x_i + b > 0$ \form#493:$3x + 5y - z = 0$ \form#494:$4x \geq 2y - 13$ \form#495:$4x > 2y - 13$ \form#496:$x - 5y + 3z \leq 4$ \form#497:$x - 5y + 3z > 4$ \form#498:$(0, 1)^\transpose \in \Rset^2$ \form#499:\[ \bigl\{\, (x, 0)^\transpose \in \Rset^2 \bigm| x \geq 0 \,\bigr\}, \] \form#500:$\vect{l} = (a_0, \ldots, a_{n-1})^\transpose$ \form#501:$\vect{r} = (a_0, \ldots, a_{n-1})^\transpose$ \form#502:$\vect{p} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$ \form#503:$\vect{c} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$ \form#504:$d > 0$ \form#505:$x-y-z$ \form#506:$\vect{p} = (1, 0, 2)^\transpose \in \Rset^3$ \form#507:$\vect{0} \in \Rset^3$ \form#508:$\vect{0} \in \Rset^2$ \form#509:$\vect{0} \in \Rset^0$ \form#510:$\vect{p}$ \form#511:$\vect{q} = (-1.5, 3.2, 2.1)^\transpose \in \Rset^3$ \form#512:$\vect{c} = (1, 0, 2)^\transpose \in \Rset^3$ \form#513:$(a_0, \ldots, a_{n-1})^\transpose$ \form#514:$(a_0, 2 a_1, \ldots, (i+1)a_i, \ldots, n a_{n-1})^\transpose$ \form#515:$x + y = 1 \pmod{2}$ \form#516:$\Zset^2$ \form#517:$\cg = \sum_{i=0}^{n-1} a_i x_i + b = 0 \pmod m$ \form#518:$m \neq 0$ \form#519:$\pmod{1}$ \form#520:$4x = 2y - 13 \pmod{1}$ \form#521:$4x = 2y - 13 \pmod{2}$ \form#522:$x - 5y + 3z = 4 \pmod{5}$ \form#523:$2 e$ \form#524:$2x - 10y + 6z = 8 \pmod{5}$ \form#525:\[ \bigl\{\, (x, 0)^\transpose \in \Rset^2 \bigm| x \pmod{1}\ 0 \,\bigr\}, \] \form#526:$x + y = 2$ \form#527:$\vect{q} = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$ \form#528:$\vect{p} = (1, -1, -1)^\transpose \in \Rset^3$ \form#529:$\vect{p1} = (-1.5, 3.2, 2.1)^\transpose \in \Rset^3$ \form#530:$\vect{q} = (1, 0, 2)^\transpose \in \Rset^3$ \form#531:$\perp$ \form#532:$M$ \form#533:$M-x'_i$ \form#534:$x'$ \form#535:$x'_i = M-x_i$ \form#536:$x_i = M-x'_i$ \form#537:$(x,y)$ \form#538:\[\left\{\begin{array}{l} y \geq 2x - 4\\ y \leq -x + p \end{array}\right.\] \form#539:\[\left\{\begin{array}{l} M - y \geq 2M - 2x - 4\\ M - y \leq -M + x + p \end{array}\right.\] \form#540:\[ \left\{\begin{array}{l} x'=M-\left\lfloor\frac{p+1}{3}\right\rfloor-1\\ y'=M-p+\left\lfloor\frac{p+1}{3}\right\rfloor+1 \end{array}\right. \] \form#541:\[ \left\{\begin{array}{l} x=\left\lfloor\frac{p+1}{3}\right\rfloor+1\\ y=p-\left\lfloor\frac{p+1}{3}\right\rfloor-1 \end{array}\right. \] \form#542:$x'_i-M$ \form#543:$x'_i = x_i+M$ \form#544:$x_i = x'_i-M$ \form#545:\[\left\{\begin{array}{l} y \geq -2x - 4\\ 2y \leq x + 2p \end{array}\right.\] \form#546:\[\left\{\begin{array}{l} y' - M \geq -2x' + 2M - 4\\ 2y' - 2M \leq x' - M + 2p \end{array}\right.\] \form#547:\[ \left\{\begin{array}{l} x'=M-\left\lfloor\frac{2p+3}{5}\right\rfloor-1\\ y'=M+2\left\lfloor\frac{2p+3}{5}\right\rfloor-2 \end{array}\right. \] \form#548:\[ \left\{\begin{array}{l} x=-\left\lfloor\frac{2p+3}{5}\right\rfloor-1\\ y=2\left\lfloor\frac{2p+3}{5}\right\rfloor-2 \end{array}\right. \] \form#549:$p^+-p^-$ \form#550:$p^+$ \form#551:$p^-$ \form#552:$p_i$ \form#553:$p^+_i-p^-$ \form#554:$-p^-$ \form#555:$x_1, x_2, \dots, x_n$ \form#556:$p_1, p_2, \dots, p_m$ \form#557:$f(x_2, \dots, x_n, p_1, \dots, p_m)$ \form#558:$x_1 \geq f(x_2, \dots, x_n, p_1, \dots, p_m)$ \form#559:$R^0$ \form#560:$\{ 2 \} \sseq \Rset$ \form#561:\[ \bigl\{\, (2, y)^\transpose \in \Rset^2 \bigm| y \in \Rset \,\bigr\}. \] \form#562:$\bigl\{ (2, 0)^\transpose \bigr\} \sseq \Rset^2$ \form#563:$x+4$ \form#564:$x+y$ \form#565:$x-y$ \form#566:$\bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2$ \form#567:$\bigl\{(3, 0)^\transpose \bigr\} \sseq \Rset^2$ \form#568:$x+3$ \form#569:$(x, y)$ \form#570:$x=y$ \form#571:$x=-y$ \form#572:$\pm x_i \leq k$ \form#573:$x_i - x_j \leq k$ \form#574:$+\infty$ \form#575:\[ a_i x_i - a_j x_j \relsym b \] \form#576:$\mathord{\relsym} \in \{ \leq, =, \geq \}$ \form#577:$a_j$ \form#578:$a_i = 0$ \form#579:$a_j = 0$ \form#580:$a_i = a_j$ \form#581:$3x - 3y \leq 1$ \form#582:$x - y \leq 1$ \form#583:$x - y \leq \frac{1}{3}$ \form#584:$x - y \leq k$ \form#585:$k > \frac{1}{3}$ \form#586:$\frac{1}{3}$ \form#587:$3x - y \leq 1$ \form#588:$\Rset^3$ \form#589:\[ ax_i + bx_j \leq k \] \form#590:$a, b \in \{-1, 0, 1\}$ \form#591:\[ \pm a_i x_i \pm a_j x_j \relsym b \] \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:$D_1$ \form#597:$D_2$ \form#598:$\fund{\gamma_1}{D_1}{\Rset^n}$ \form#599:$\fund{\gamma_2}{D_2}{\Rset^n}$ \form#600:$D = D_1 \times D_2$ \form#601:$\fund{\gamma}{D}{\Rset^n}$ \form#602:$d = (d_1, d_2) \in D$ \form#603:\[ \gamma(d) = \gamma_1(d_1) \inters \gamma_2(d_2). \] \form#604:$d = (G, P) \in (\Gset \times \Pset)$ \form#605:$G$ \form#606:$-\infty$ \form#607:$ y \cdot 2^\mathtt{exp} $ \form#608:$ y / 2^\mathtt{exp} $ \form#609:$15 \cdot 10^2 = 1500$ \form#610:$5$ \form#611:$3/580000$ \form#612:$713/10$ \form#613:$-1929/15625$ \form#614:$11/50$ \form#615:$-220001/100$ \form#616:$35$ \form#617:$44027$ \form#618:$7/2$ \form#619:$2000$ \form#620:$1024$ \form#621:$1088$ \form#622:$9073863231288$ \form#623:$17/8$ \form#624:$482$ \form#625:$256$ \form#626:$\epsilon$ \form#627:$a_n$ \form#628:$c = \bigl(\sum_{i=0}^{n-1} a_i x_i + b \relsym 0\bigr)$ \form#629:$\sum_{i=0}^{n-1} a_i x_i + b$ \form#630:$g = (\frac{a_0}{d}, \ldots, \frac{a_{n-1}}{d})^\transpose$ \form#631:$d = 1$ \form#632:$\sum_{i=0}^{n-1} a_i x_i$ \form#633:$cg = \bigl(\sum_{i=0}^{n-1} a_i x_i + b = 0 \pmod{m}\bigr)$ \form#634:$r \times c$ \form#635:$(r+n) \times c$ \form#636:$\genfrac{(}{)}{0pt}{}{M}{0}$ \form#637:$r \times (c+n)$ \form#638:$(M \, 0)$ \form#639:$(r+n) \times (c+m)$ \form#640:$\bigl(\genfrac{}{}{0pt}{}{M}{0} \genfrac{}{}{0pt}{}{0}{0}\bigr)$ \form#641:$(r+1) \times c$ \form#642:$\genfrac{(}{)}{0pt}{}{M}{y}$ \form#643:$\times$ \form#644:$M \in \Rset^r \times \Rset^c$ \form#645:$N \in \Rset^{r+n} \times \Rset^{c+n}$ \form#646:$N = \bigl(\genfrac{}{}{0pt}{}{0}{M}\genfrac{}{}{0pt}{}{J}{o}\bigr)$ \form#647:$n \times n$ \form#648:$\sum_{i = 0}^{n - 1} a_i x_i + b$ \form#649:${a'}_{ij}$ \form#650:$a_{ij}$ \form#651:\[ {a'}_{ij} = \begin{cases} a_{ij} * \mathrm{denominator} + a_{iv} * \mathrm{expr}[j] \quad \text{for } j \neq v; \\ \mathrm{expr}[v] * a_{iv} \quad \text{for } j = v. \end{cases} \] \form#652:$0 \leq 1$ \form#653:$\sum_{i=0}^{n-1} 0 x_i + 0 = 0$ \form#654:$\sum_{i=0}^{n-1} 0 x_i + b \geq 0$ \form#655:$b \geq 0$ \form#656:$\sum_{i=0}^{n-1} 0 x_i + b > 0$ \form#657:$b > 0$ \form#658:$\sum_{i=0}^{n-1} 0 x_i + b = 0$ \form#659:$b \neq 0$ \form#660:$b < 0$ \form#661:$b \leq 0$ \form#662:$\epsilon \geq 0$ \form#663:$\epsilon \leq 1$ \form#664:\[ \frac{\sum_{i = 0}^{n - 1} a_i x_i + b} {\mathrm{denominator}}. \] \form#665:$L_\infty$ \form#666:$ e_1 = e_2 $ \form#667:$ e_1 = e_2 \pmod{0}$ \form#668:$(r+dims) \times (c+dims)$ \form#669:$\bigl(\genfrac{}{}{0pt}{}{0}{A} \genfrac{}{}{0pt}{}{B}{A}\bigr)$ \form#670:$B$ \form#671:$dims \times dims$ \form#672:$\bigl(\genfrac{}{}{0pt}{}{0}{1} \genfrac{}{}{0pt}{}{1}{0}\bigr)$ \form#673:$e1 = e2 \pmod{1}$ \form#674:$e = n \pmod{1}$ \form#675:$ e_1 = e_2 \pmod{m}$ \form#676:$ e_1 = e_2 \pmod{mk}$ \form#677:$\sum_{i=0}^{n-1} 0 x_i + 0 == 0$ \form#678:$\sum_{i=0}^{n-1} 0 x_i + b \%= 0 / m$ \form#679:$b = 0 \pmod{m}$ \form#680:$\sum_{i=0}^{n-1} 0 x_i + b == 0$ \form#681:$b \neq 0 \pmod{m}$ \form#682:$0 = 1 \pmod{1}$ \form#683:$0 = 1 \pmod{0}$ \form#684:$n = e \pmod{1}$ \form#685:$\bigl(\genfrac{}{}{0pt}{}{A}{0} \genfrac{}{}{0pt}{}{0}{B}\bigr)$ \form#686:$\bigl(\genfrac{}{}{0pt}{}{1}{0} \genfrac{}{}{0pt}{}{0}{1}\bigr)$ \form#687:$[\mathrm{first}, \mathrm{last})$ \form#688:$\frac{\mathit{expr}}{\mathit{den}}$ \form#689:$f_i(x,p) = 0 ; 1 \leq i \leq n$ \form#690:$f_i(x,p) \geq 0 ; 1 \leq i \leq n$ \form#691:$\sum\limits_{i=1}^n f_i(x,p) \leq 0$ \form#692:$c=s_{*j}\frac{t_{ik}}{s_{ij}}$ \form#693:$c'=s_{*j'}\frac{t_{i'k}}{s_{i'j'}}$ \form#694:$s_{*j}$ \form#695:$s$ \form#696:$s_{*j'}$ \form#697:$j'$ \form#698:$(i,j)$ \form#699:$c'$ \form#700:$(i',j')$ \form#701:$-c$ \form#702:$-c'$ \form#703:$\mathrm{var}' \relsym \frac{\mathrm{expr}}{\mathrm{denominator}}$ \form#704:$\mathrm{lhs}' \relsym \mathrm{rhs}$ \form#705:$\frac{\mathrm{lb\_expr}}{\mathrm{denominator}} \leq \mathrm{var}' \leq \frac{\mathrm{ub\_expr}}{\mathrm{denominator}}$ \form#706:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cP \,\bigr\}. \] \form#707:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cP \,\bigr\}. \] \form#708:$f(k)$ \form#709:$\dots$ \form#710:$\Rset^0 = \{\cdot\}$ \form#711:\[ \begin{cases} sat\_c[i][j] = 0, \quad \text{if } G[i] \cdot C^\mathrm{T}[j] = 0; \\ sat\_c[i][j] = 1, \quad \text{if } G[i] \cdot C^\mathrm{T}[j] > 0. \end{cases} \] \form#712:\[ \begin{cases} sat\_g[i][j] = 0, \quad \text{if } C[i] \cdot G^\mathrm{T}[j] = 0; \\ sat\_g[i][j] = 1, \quad \text{if } C[i] \cdot G^\mathrm{T}[j] > 0. \end{cases} \] \form#713:$\frac{num}{den}$ \form#714:$\frac{d_{j}}{\|\Delta x^{j} \|}$ \form#715:\[ \|\Delta x^{j} \| = \left( 1+\sum_{i=1}^{m} \alpha_{ij}^2 \right)^{\frac{1}{2}}. \] \form#716:$\alpha$ \form#717:\[ \{\, a \in \mathtt{to} \mid \exists b \in \mathtt{x} \st a \mathrel{\mathtt{rel}} b \,\}. \] \form#718:\[ \{\, a \in \mathtt{to} \mid \forall b \in \mathtt{x} \itc a \mathrel{\mathtt{rel}} b \,\}. \] \form#719:$\mathrm{var}' = \frac{\mathrm{expr}}{\mathrm{denominator}} \pmod{\mathrm{modulus}}$ \form#720:$\mathrm{lhs}' = \mathrm{rhs} \pmod{\mathrm{modulus}}$ \form#721:$\cL \sseq \Rset^2$ \form#722:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cL \,\bigr\}. \] \form#723:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cL \,\bigr\}. \] \form#724:$\{3x \equiv_3 0, 4x + y \equiv_3 1\}$ \form#725:$\{3x \equiv_3 0, x + y \equiv_3 1\}$ \form#726:$\mathtt{y} \Delta \mathtt{x}$ \form#727:$\cB \sseq \Rset^2$ \form#728:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cB \,\bigr\}. \] \form#729:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cB \,\bigr\}. \] \form#730:$B \sseq \Rset^n$ \form#731:$D \sseq \Rset^m$ \form#732:$R \sseq \Rset^{n+m}$ \form#733:\[ R \defeq \Bigl\{\, (x_1, \ldots, x_n, y_1, \ldots, y_m)^\transpose \Bigm| (x_1, \ldots, x_n)^\transpose \in B, (y_1, \ldots, y_m)^\transpose \in D \,\Bigl\}. \] \form#734:$I$ \form#735:$d$ \form#736:$n/d$ \form#737:$0/1$ \form#738:$\mathrm{var} \relsym \frac{\mathrm{expr}}{\mathrm{denominator}}$ \form#739:$\cO \sseq \Rset^2$ \form#740:\[ \bigl\{\, (x, y, z)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cO \,\bigr\}. \] \form#741:\[ \bigl\{\, (x, y, 0)^\transpose \in \Rset^3 \bigm| (x, y)^\transpose \in \cO \,\bigr\}. \] \form#742:$O(n^2)$ \form#743:\[ \mathit{ps} = \bigl\{ \{ x \geq 0 \}, \{ x \leq 0 \} \bigr\}, \] \form#744:$\mathit{ps}$ \form#745:$\cP \uplus \cQ \neq \cP \union \cQ$ \form#746:$ x'_1, \ldots, x'_n $ \form#747:$ 0, \ldots, n-1 $ \form#748:$ x_1, \ldots, x_n $ \form#749:$ n, \ldots, 2n-1 $ \form#750:$ n+1 $ \form#751:$ \mu_0 + \sum_{i=1}^n \mu_i x_i $ \form#752:$ \mu_0, \mu_1, \ldots, \mu_n $ \form#753:$ n, 0, \ldots, n-1 $