Lines Matching refs:ge
196 $A \vec x + \vec b \ge \vec 0$, the tableau represents the relationship
286 value of $r$ and $\vec c \ge 0$, the expression is non-negative
392 $2 x - 5 \ge 0$ should be simplified to $x - 3 \ge 0$.
452 $b_i(\vec p) + \sp {\vec a} {\vec x} \ge 0$,
460 constraint $u + \sp {\vec a} {\vec x} \ge 0$ along with
464 $\sp {\vec a} {\vec x} \ge -u \ge -b_i(\vec p)$.
541 $\min_i b_i(\vec p) \ge f_j(\vec p)$, then each of these constraints
782 $b_i(\vec p) + \sp {\vec a} {\vec x} \ge 0$ constraints,
801 That is, $r = b(\vec p) + c_j - f$ with $f = -\sum_{i\ne j} a_i c_i \ge 0$.
803 context constraints $t \ge -b(\vec p)$ and $t \ge 0$ and replace
807 by $t \ge -b(\vec p)$ while and non-negative value remains non-negative
808 because $t \ge 0$.
814 If $b(\vec p) \ge 0$, we can take $t = 0$ so that
815 $c' = c_j - t = c_j \ge 0$.
817 Since $r = b(\vec p) + c_j - f \ge 0$ and $f \ge 0$, we have
818 $c' = c_j + b(\vec p) \ge 0$.
841 $k \in \Z_{\ge 1}$
849 R \circ R^{k-1} & \text{if $k \ge 2$}
860 R^+ \coloneqq \bigcup_{k \ge 1} R^k
886 and first compute an approximation of $R^k$ for $k \ge 1$ and then project
894 R^k = k \mapsto \{\, x \to x + k \mid k \ge 1 \,\}
900 R^+ & = \{\, x \to y \mid \exists k \in \Z_{\ge 1} : y = x + k \,\}
902 & = \{\, x \to y \mid y \ge x + 1 \,\}
939 \exists k_i \in \Z_{\ge 0}, \vec\delta_i \in k_i \, \Delta_i(\vec s) :
959 \exists k_i \in \Z_{\ge 0} :
994 \exists k \in \Z_{\ge 1}, \vec \delta \in k \, \Delta_i'(\vec s) :
1065 \mid {} & \exists k \in \Z_{\ge 1}, \vec f \in \Z^d :
1081 $k \in \Z_{\ge 1}$ and for every $\vec f \in k \, \Delta_i(\vec s)$
1101 \ge \vec 0
1109 i.e., $A_3 \vec f_j \ge \vec r(\vec s) \ge \vec 0$
1110 and therefore also $A_3 \vec f \ge \vec r(\vec s)$.
1113 $\{\, \vec x \in \Q^d \mid A_1 \vec x + \vec c_1 \ge \vec 0\,\}$,
1127 $k \, \Delta_i(\vec s)$ with $k \ge 1$.
1135 $\spv a x + \spv b s + c \ge 0$ valid for $\Delta_i(\vec s)$.
1137 $k \, \spv a x + \spv b s + c \ge 0$.
1144 $\vec b$ and $c$ need to be such that $- \spv b s - c \ge 0$ is a valid
1160 n \to \{\, (x, y) \to (1 + x, 1 - n + y) \mid n \ge 2 \,\}
1165 \Delta = n \to \{\, (1, 1 - n) \mid n \ge 2 \,\}
1169 $y - 1 + n \ge 0$, leading to the following approximation of the
1172 n \to \{\, (x, y) \to (o_0, o_1) \mid n \ge 2 \wedge o_1 \le 1 - n + y \wedge o_0 \ge 1 + x \,\}
1242 …sts \, \alpha_0, \alpha_1: 7\alpha_0 = -2 + n \wedge 5\alpha_1 = -1 - x + y \wedge y \ge 6 + x \,\}
1248 …mid \exists \, \alpha_0, \alpha_1: 7\alpha_0 = -2 + n \wedge 5\alpha_1 = -1 + x \wedge x \ge 6 \,\}
1271 k \ge 1 \wedge
1277 x \ge 6 k
1286 n \to \{\, x \to y \mid \exists \, \alpha_0, \alpha_1: 7\alpha_1 = -2 + n \wedge 6\alpha_0 \ge -x +…
1322 R = n \to \{\, (x, y) \to (1 + x, 1 - n + y) \mid n \ge 2 \,\}
1327 \diff R = n \to \{\, (1, 1 - n) \mid n \ge 2 \,\}
1332 n \to \{\, (x, y) \to (x', y') \mid n \ge 2 \wedge y' \le 1 - n + y \wedge x' \ge 1 + x \,\}
1337 R' = \{\, (n, x, y) \to (n, 1 + x, 1 - n + y) \mid n \ge 2 \,\}
1345 n \to \{\, (x, y) \to (x', y') \mid n \ge 2 \wedge x' \ge 1 + x \wedge y' \le x + y - x' \,\}
1350 n \to \{\, (x, y) \to (x', y') \mid n \ge 2 \wedge y' \le 1 - n + y \wedge x' \ge 1 + x \wedge y' \…
1356 n \to \{\, (x, y) \to (x', y') \mid y' \le 2 - n + x + y - x' \wedge n \ge 2 \wedge x' \ge 1 + x \,…
1400 K'|_{y_{d+1} - x_{d+1} \ge 2} & \subseteq R' \circ K'|_{y_{d+1} - x_{d+1} \ge 1}
1763 \{\, (x, y) \to (x_2, y_2) \mid {} & (3y = 2x \wedge x_2 = x \wedge 3y_2 = 3 + 2x \wedge x \ge 0 \w…
1764 & (x_2 = 1 + x \wedge y_2 = y \wedge x \ge 0 \wedge 3y \ge 2 + 2x \wedge x \le 2 \wedge 3y \le 3 + …
1782 (x \ge 0 \wedge 3y \ge 2 + 2x \wedge x \le 2 \wedge 3y \le 3 + 2x) \,\}
1785 \{\, (x, y) \to (x_2, y_2) \mid (3y = 2x \wedge x_2 = x \wedge 3y_2 = 3 + 2x \wedge x \ge 0 \wedge …
1845 i.e., the part that results from the paths of positive length ($k \ge 1$),
1998 k \ge 1 \wedge
2026 k \ge 0 \wedge