SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} + \frac {8 3 8 8 6 0 8 L ^ {3} \hat {L} ^ {2} \omega p ^ {4}}{\bar {L} ^ {5} \alpha^ {3} \beta^ {2} n \tau^ {2}} + \frac {8 3 8 8 6 0 8 L \hat {L} ^ {4} \omega^ {2} p ^ {2} \tau}{\bar {L} ^ {5} \alpha^ {3} \beta n ^ {2}} + \frac {8 3 8 8 6 0 8 L \hat {L} ^ {4} \omega^ {2} p ^ {3} \tau}{\bar {L} ^ {5} \alpha^ {3} \beta^ {2} n ^ {2}} \\ + \frac {6 7 1 0 8 8 6 4 L \hat {L} ^ {4} \omega^ {2} p ^ {4} \tau}{\bar {L} ^ {5} \alpha^ {3} \beta^ {4} n ^ {2}} + \frac {6 7 1 0 8 8 6 4 L \hat {L} ^ {4} \omega^ {2} p ^ {2} \tau}{\bar {L} ^ {5} \alpha^ {3} \beta^ {2} n ^ {2}} + \frac {1 3 4 2 1 7 7 2 8 L \hat {L} ^ {4} \omega^ {2} p ^ {3} \tau}{\bar {L} ^ {5} \alpha^ {3} \beta^ {3} n ^ {2}} \\ + \frac {2 6 8 4 3 5 4 5 6 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {5}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {4} n ^ {2} \tau} + \frac {2 6 8 4 3 5 4 5 6 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {3}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {2} n ^ {2} \tau} + \frac {5 3 6 8 7 0 9 1 2 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {6}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {4} n ^ {2} \tau^ {2}} \\ + \frac {5 3 6 8 7 0 9 1 2 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {4}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {3} n ^ {2} \tau} + \frac {5 3 6 8 7 0 9 1 2 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {4}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {2} n ^ {2} \tau^ {2}} + \frac {1 0 7 3 7 4 1 8 2 4 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {5}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {3} n ^ {2} \tau^ {2}} \\ + \frac {1 0 7 3 7 4 1 8 2 4 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {5}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {4} n ^ {2} \tau} + \frac {1 0 7 3 7 4 1 8 2 4 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {3}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {2} n ^ {2} \tau} + \frac {2 1 4 7 4 8 3 6 4 8 L \hat {L} ^ {6} \omega^ {3} p ^ {5} \tau}{\bar {L} ^ {7} \alpha^ {4} \beta^ {4} n ^ {3}} \\ + \frac {2 1 4 7 4 8 3 6 4 8 L \hat {L} ^ {6} \omega^ {3} p ^ {3} \tau}{\bar {L} ^ {7} \alpha^ {4} \beta^ {2} n ^ {3}} + \frac {2 1 4 7 4 8 3 6 4 8 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {6}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {4} n ^ {2} \tau^ {2}} + \frac {2 1 4 7 4 8 3 6 4 8 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {4}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {3} n ^ {2} \tau} \\ + \frac {2 1 4 7 4 8 3 6 4 8 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {4}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {2} n ^ {2} \tau^ {2}} + \frac {4 2 9 4 9 6 7 2 9 6 L \hat {L} ^ {6} \omega^ {3} p ^ {4} \tau}{\bar {L} ^ {7} \alpha^ {4} \beta^ {3} n ^ {3}} + \frac {4 2 9 4 9 6 7 2 9 6 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {5}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {3} n ^ {2} \tau^ {2}} \leq 1 \\ \end{array}
$$
N Symbolically Computed Constraints for  $\bar{L}$  Such That The Inequality from Section M Holds
$$
\frac {2 L}{\bar {L}} \leq \frac {1}{1 1 4} \quad (2 7 2) \quad \frac {1 0 2 4 L ^ {3}}{\bar {L} ^ {3} \alpha^ {2}} \leq \frac {1}{1 1 4} \quad (2 7 3) \quad \frac {4 0 9 6 L ^ {3} p}{\bar {L} ^ {3} \alpha^ {2} \tau} \leq \frac {1}{1 1 4} \tag {274}
$$
$$
\frac {8 1 9 2 L ^ {3} p ^ {2}}{\bar {L} ^ {3} \alpha^ {2} \tau^ {2}} \leq \frac {1}{1 1 4} \quad (2 7 5) \quad \frac {2 5 6 \hat {L} ^ {2} \omega p}{\bar {L} ^ {2} \beta n} \leq \frac {1}{1 1 4} \quad (2 7 6) \quad \frac {2 5 6 \hat {L} ^ {2} \omega p ^ {2}}{\bar {L} ^ {2} \beta^ {2} n} \leq \frac {1}{1 1 4} \tag {277}
$$
$$
\frac {1 0 2 4 L \hat {L} ^ {2} \omega p}{\bar {L} ^ {3} \alpha \beta n} \leq \frac {1}{1 1 4} \quad (2 7 8) \quad \frac {1 0 2 4 L \hat {L} ^ {2} \omega p ^ {2}}{\bar {L} ^ {3} \alpha \beta^ {2} n} \leq \frac {1}{1 1 4} \quad (2 7 9) \quad \frac {8 1 9 2 L \hat {L} ^ {2} \omega p \tau}{\bar {L} ^ {3} \alpha^ {2} n} \leq \frac {1}{1 1 4} \tag {280}
$$
$$
\frac {1 3 1 0 7 2 L ^ {2} \hat {L} ^ {2} \omega p}{\bar {L} ^ {4} \alpha^ {2} \beta n} \leq \frac {1}{1 1 4} \quad (2 8 1) \quad \frac {1 3 1 0 7 2 L ^ {2} \hat {L} ^ {2} \omega p ^ {2}}{\bar {L} ^ {4} \alpha^ {2} \beta^ {2} n} \leq \frac {1}{1 1 4} \quad (2 8 2) \quad \frac {1 0 4 8 5 7 6 L ^ {3} \hat {L} ^ {2} \omega p}{\bar {L} ^ {5} \alpha^ {3} \beta n} \leq \frac {1}{1 1 4} \tag {283}
$$
$$
\frac {1 0 4 8 5 7 6 L ^ {3} \hat {L} ^ {2} \omega p ^ {2}}{\bar {L} ^ {5} \alpha^ {3} \beta^ {2} n} \leq \frac {1}{1 1 4} \quad (2 8 4) \quad \frac {1 0 4 8 5 7 6 \hat {L} ^ {4} \omega^ {2} p ^ {2} \tau}{\bar {L} ^ {4} \alpha^ {2} \beta n ^ {2}} \leq \frac {1}{1 1 4} \quad (2 8 5) \quad \frac {1 0 4 8 5 7 6 \hat {L} ^ {4} \omega^ {2} p ^ {3} \tau}{\bar {L} ^ {4} \alpha^ {2} \beta^ {2} n ^ {2}} \leq \frac {1}{1 1 4} \quad (2 8 6)
$$
$$
\frac {1 6 7 7 7 2 1 6 \hat {L} ^ {4} \omega^ {2} p ^ {4} \tau}{\bar {L} ^ {4} \alpha^ {2} \beta^ {4} n ^ {2}} \leq \frac {1}{1 1 4} (2 8 7) \quad \frac {1 6 7 7 7 2 1 6 \hat {L} ^ {4} \omega^ {2} p ^ {2} \tau}{\bar {L} ^ {4} \alpha^ {2} \beta^ {2} n ^ {2}} \leq \frac {1}{1 1 4} (2 8 8) \quad \frac {3 3 5 5 4 4 3 2 \hat {L} ^ {4} \omega^ {2} p ^ {3} \tau}{\bar {L} ^ {4} \alpha^ {2} \beta^ {3} n ^ {2}} \leq \frac {1}{1 1 4} (2 8 9)
$$
$$
\frac {6 7 1 0 8 8 6 4 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {4}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {4} n ^ {2}} \leq \frac {1}{1 1 4} \quad \frac {6 7 1 0 8 8 6 4 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {2}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {2} n ^ {2}} \leq \frac {1}{1 1 4} \quad \frac {1 3 4 2 1 7 7 2 8 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {3}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {3} n ^ {2}} \leq \frac {1}{1 1 4} \tag {292}
$$
$$
\frac {2 6 8 4 3 5 4 5 6 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {4}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {4} n ^ {2}} \leq \frac {1}{1 1 4} \tag {293}
$$
$$
\frac {2 6 8 4 3 5 4 5 6 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {2}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {2} n ^ {2}} \leq \frac {1}{1 1 4} \tag {294}
$$
$$
\frac {5 3 6 8 7 0 9 1 2 \hat {L} ^ {6} \omega^ {3} p ^ {5} \tau}{\bar {L} ^ {6} \alpha^ {3} \beta^ {4} n ^ {3}} \leq \frac {1}{1 1 4} \tag {295}
$$
$$
\frac {5 3 6 8 7 0 9 1 2 \hat {L} ^ {6} \omega^ {3} p ^ {3} \tau}{\bar {L} ^ {6} \alpha^ {3} \beta^ {2} n ^ {3}} \leq \frac {1}{1 1 4} \tag {296}
$$
$$
\frac {5 3 6 8 7 0 9 1 2 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {3}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {3} n ^ {2}} \leq \frac {1}{1 1 4} \tag {297}
$$
$$
\frac {1 0 7 3 7 4 1 8 2 4 \hat {L} ^ {6} \omega^ {3} p ^ {4} \tau}{\bar {L} ^ {6} \alpha^ {3} \beta^ {3} n ^ {3}} \leq \frac {1}{1 1 4} \tag {298}
$$
$$
\frac {1 3 1 0 7 2 L \hat {L} ^ {2} \omega p \tau}{\bar {L} ^ {3} \alpha^ {2} \beta n} \leq \frac {1}{1 1 4} \tag {299}
$$
$$
\frac {1 3 1 0 7 2 L \hat {L} ^ {2} \omega p ^ {2} \tau}{\bar {L} ^ {3} \alpha^ {2} \beta^ {2} n} \leq \frac {1}{1 1 4} \quad (3 0 0)
$$
$$
\frac {5 2 4 2 8 8 L ^ {2} \hat {L} ^ {2} \omega p ^ {2}}{\bar {L} ^ {4} \alpha^ {2} \beta n \tau} \leq \frac {1}{1 1 4} \tag {301}
$$
$$
\frac {5 2 4 2 8 8 L ^ {2} \hat {L} ^ {2} \omega p ^ {3}}{\bar {L} ^ {4} \alpha^ {2} \beta^ {2} n \tau} \leq \frac {1}{1 1 4} \tag {302}
$$
$$
\frac {1 0 4 8 5 7 6 L ^ {2} \hat {L} ^ {2} \omega p ^ {3}}{\bar {L} ^ {4} \alpha^ {2} \beta n \tau^ {2}} \leq \frac {1}{1 1 4} \tag {303}
$$
$$
\frac {1 0 4 8 5 7 6 L ^ {2} \hat {L} ^ {2} \omega p ^ {4}}{\bar {L} ^ {4} \alpha^ {2} \beta^ {2} n \tau^ {2}} \leq \frac {1}{1 1 4} \tag {304}
$$
$$
\frac {4 1 9 4 3 0 4 L ^ {3} \hat {L} ^ {2} \omega p ^ {2}}{\bar {L} ^ {5} \alpha^ {3} \beta n \tau} \leq \frac {1}{1 1 4} \tag {305}
$$
$$
\frac {4 1 9 4 3 0 4 L ^ {3} \hat {L} ^ {2} \omega p ^ {3}}{\bar {L} ^ {5} \alpha^ {3} \beta^ {2} n \tau} \leq \frac {1}{1 1 4} \quad (3 0 6)
$$
$$
\frac {8 3 8 8 6 0 8 L ^ {3} \hat {L} ^ {2} \omega p ^ {3}}{\bar {L} ^ {5} \alpha^ {3} \beta n \tau^ {2}} \leq \frac {1}{1 1 4} \tag {307}
$$
$$
\frac {8 3 8 8 6 0 8 L ^ {3} \hat {L} ^ {2} \omega p ^ {4}}{\bar {L} ^ {5} \alpha^ {3} \beta^ {2} n \tau^ {2}} \leq \frac {1}{1 1 4} \tag {308}
$$
$$
\frac {8 3 8 8 6 0 8 L \hat {L} ^ {4} \omega^ {2} p ^ {2} \tau}{\bar {L} ^ {5} \alpha^ {3} \beta n ^ {2}} \leq \frac {1}{1 1 4} \tag {309}
$$
$$
\frac {8 3 8 8 6 0 8 L \hat {L} ^ {4} \omega^ {2} p ^ {3} \tau}{\bar {L} ^ {5} \alpha^ {3} \beta^ {2} n ^ {2}} \leq \frac {1}{1 1 4} \tag {310}
$$
$$
\frac {6 7 1 0 8 8 6 4 L \hat {L} ^ {4} \omega^ {2} p ^ {4} \tau}{\bar {L} ^ {5} \alpha^ {3} \beta^ {4} n ^ {2}} \leq \frac {1}{1 1 4} \tag {311}
$$
$$
\frac {6 7 1 0 8 8 6 4 L \hat {L} ^ {4} \omega^ {2} p ^ {2} \tau}{\bar {L} ^ {5} \alpha^ {3} \beta^ {2} n ^ {2}} \leq \frac {1}{1 1 4} \tag {312}
$$
$$
\frac {1 3 4 2 1 7 7 2 8 L \hat {L} ^ {4} \omega^ {2} p ^ {3} \tau}{\bar {L} ^ {5} \alpha^ {3} \beta^ {3} n ^ {2}} \leq \frac {1}{1 1 4} \tag {313}
$$
$$
\frac {2 6 8 4 3 5 4 5 6 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {5}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {4} n ^ {2} \tau} \leq \frac {1}{1 1 4} \tag {314}
$$
$$
\frac {2 6 8 4 3 5 4 5 6 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {3}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {2} n ^ {2} \tau} \leq \frac {1}{1 1 4} \tag {315}
$$
$$
\frac {5 3 6 8 7 0 9 1 2 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {6}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {4} n ^ {2} \tau^ {2}} \leq \frac {1}{1 1 4} \tag {316}
$$
$$
\frac {5 3 6 8 7 0 9 1 2 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {4}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {3} n ^ {2} \tau} \leq \frac {1}{1 1 4} \tag {317}
$$
$$
\frac {5 3 6 8 7 0 9 1 2 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {4}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {2} n ^ {2} \tau^ {2}} \leq \frac {1}{1 1 4} \tag {318}
$$
$$
\frac {1 0 7 3 7 4 1 8 2 4 L ^ {2} \hat {L} ^ {4} \omega^ {2} p ^ {5}}{\bar {L} ^ {6} \alpha^ {3} \beta^ {3} n ^ {2} \tau^ {2}} \leq \frac {1}{1 1 4} \tag {319}
$$
$$
\frac {1 0 7 3 7 4 1 8 2 4 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {5}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {4} n ^ {2} \tau} \leq \frac {1}{1 1 4} \tag {320}
$$
$$
\frac {1 0 7 3 7 4 1 8 2 4 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {3}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {2} n ^ {2} \tau} \leq \frac {1}{1 1 4} \tag {321}
$$
$$
\frac {2 1 4 7 4 8 3 6 4 8 L \hat {L} ^ {6} \omega^ {3} p ^ {5} \tau}{\bar {L} ^ {7} \alpha^ {4} \beta^ {4} n ^ {3}} \leq \frac {1}{1 1 4} \tag {322}
$$
$$
\frac {2 1 4 7 4 8 3 6 4 8 L \hat {L} ^ {6} \omega^ {3} p ^ {3} \tau}{\bar {L} ^ {7} \alpha^ {4} \beta^ {2} n ^ {3}} \leq \frac {1}{1 1 4} \quad \frac {2 1 4 7 4 8 3 6 4 8 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {6}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {4} n ^ {2} \tau^ {2}} \leq \frac {1}{1 1 4} \quad \frac {2 1 4 7 4 8 3 6 4 8 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {4}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {3} n ^ {2} \tau} \leq \frac {1}{1 1 4} \tag {325}
$$
$$
\frac {2 1 4 7 4 8 3 6 4 8 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {4}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {2} n ^ {2} \tau^ {2}} \leq \frac {1}{1 1 4} \quad \frac {4 2 9 4 9 6 7 2 9 6 L \hat {L} ^ {6} \omega^ {3} p ^ {4} \tau}{\bar {L} ^ {7} \alpha^ {4} \beta^ {3} n ^ {3}} \leq \frac {1}{1 1 4} \quad \frac {4 2 9 4 9 6 7 2 9 6 L ^ {3} \hat {L} ^ {4} \omega^ {2} p ^ {5}}{\bar {L} ^ {7} \alpha^ {4} \beta^ {3} n ^ {2} \tau^ {2}} \leq \frac {1}{1 1 4} \tag {328}
$$
O Symbolical Check That The Constraints from Sections I, J, L and N Follow From The Constraint (44)
Note that the inequalities from Lemma F.1 follow from (44). Therefore, the inequalities from Sections I, J, L and N follow from (44), if they follow from the inequalities from Lemma F.1. We now present it<sup>7</sup>. These results are checked and generated using the script in Section P (see Section 6 in Section P)
(93) follows from (74), (74).
(94) follows from (73), (73).
(95) follows from (77), (77).
(96) follows from (78), (77).
(97) follows from (78), (78).
(98) follows from (72), (72).
(99) follows from (71), (71).
(100) follows from (70), (70).
(101) follows from (77), (77), (74), (74).
(102) follows from (77), (77), (73), (73).
(103) follows from (74), (74), (72), (72).
(104) follows from (73), (73), (72), (72).
(105) follows from (78), (77), (74), (74).
(106) follows from (78), (77), (73), (73).
(107) follows from (78), (78), (74), (74).
(108) follows from (78), (78), (73), (73).
(109) follows from (77), (77).
(110) follows from (77), (77), (77), (77).
(111) follows from (66).
(112) follows from (78), (77).
(113) follows from (78), (78).
(114) follows from (78), (77), (77), (77).
(115) follows from (78), (78), (77), (77).
(116) follows from (77), (66).
(117) follows from (65).
(118) follows from (64).
(119) follows from (77), (77), (77), (66).
$7^{\prime \prime}(a)$  follows from  $(b),(c)$  means that one should use  $(b)$  and  $(c)$  to get  $(a)$ . It is possible that  $b = c$ , thus one should apply  $b(= c)$  two times.
(120) follows from (77), (65).
(121) follows from (77), (64).
(122) follows from (74), (74), (66).
(123) follows from (74), (74), (65).
(124) follows from (69), (69).
(125) follows from (78), (77), (77), (66).
(126) follows from (78), (78), (77), (66).
(127) follows from (77), (74), (74).
(128) follows from (77), (73), (73).
(129) follows from (77), (77), (72), (72).
(130) follows from (77), (77), (77), (65).
(131) follows from (77), (77), (77), (64).
(132) follows from (77), (77), (74), (74).
(133) follows from (77), (77), (73), (73).
(134) follows from (77), (77), (77), (74), (74).
(135) follows from (77), (77), (77), (73), (73).
(136) follows from (77), (77), (77), (77), (74), (74).
(137) follows from (77), (77), (77), (77), (73), (73).
(138) follows from (77), (77), (77), (73), (73), (73), (73).
(139) follows from (77), (77), (77), (74), (74), (74), (74).
(140) follows from (77), (77), (77), (74), (74), (73), (73).
(141) follows from (77), (77), (77), (77), (73), (73), (73), (73).
(142) follows from (77), (77), (77), (77), (74), (74), (74), (74).
(143) follows from (77), (77), (77), (77), (74), (74), (73), (73).
(144) follows from (77), (74), (74), (66).
(145) follows from (77), (74), (74), (65).
(146) follows from (68), (68).
(147) follows from (67), (67).
(148) follows from (77), (72), (72), (66).
(149) follows from (78), (74), (74).
(150) follows from (78), (73), (73).
(151) follows from (81), (81), (81).
(152) follows from (80), (80), (80).
(153) follows from (72), (72), (65).
(154) follows from (72), (72), (64).
(155) follows from (78), (77), (77), (65).
(156) follows from (78), (77), (77), (64).
(157) follows from (78), (77), (74), (74).
(158) follows from (78), (77), (73), (73).
(159) follows from (77), (77), (74), (74), (66).
(160) follows from (78), (78), (77), (65).
(161) follows from (78), (78), (77), (64).
(162) follows from (77), (77), (71), (71).
(163) follows from (77), (77), (70), (70).
(164) follows from (78), (78), (74), (74).
(165) follows from (78), (78), (73), (73).
(166) follows from (78), (77), (77), (74), (74).
(167) follows from (78), (77), (77), (73), (73).
(168) follows from (77), (77), (77), (74), (74), (66).
(169) follows from (70), (70), (64).
(170) follows from (71), (71), (65).
(171) follows from (74), (74), (72), (72), (66).
(172) follows from (78), (78), (77), (74), (74).
(173) follows from (78), (78), (77), (73), (73).
(174) follows from (77), (74), (74), (72), (72).
(175) follows from (77), (73), (73), (72), (72).
(176) follows from (71), (71), (64).
(177) follows from (77), (77), (73), (73), (64).
(178) follows from (77), (77), (74), (74), (65).
(179) follows from (78), (77), (77), (77), (74), (74).
(180) follows from (78), (77), (77), (77), (73), (73).
(181) follows from (77), (77), (74), (74), (64).
(182) follows from (78), (78), (77), (77), (74), (74).
(183) follows from (78), (78), (77), (77), (73), (73).
(184) follows from (74), (74), (71), (71), (64).
(185) follows from (74), (74), (71), (71), (66).
(186) follows from (77), (77), (77), (73), (73), (64).
(187) follows from (77), (77), (74), (74), (72), (72).
(188) follows from (79), (78), (71), (71), (71), (71).
(189) follows from (77), (77), (77), (74), (74), (65).
(190) follows from (77), (73), (73), (70), (70).
(191) follows from (77), (74), (74), (71), (71).
(192) follows from (73), (73), (72), (72), (64).
(193) follows from (74), (74), (71), (71), (65).
(194) follows from (77), (77), (77), (74), (74), (64).
(195) follows from (74), (74), (72), (72), (65).
(196) follows from (77), (74), (74), (70), (70).
(197) follows from (74), (74), (72), (72), (64).
(198) follows from (77), (77), (74), (74), (74), (74), (64).
(199) follows from (77), (77), (74), (74), (74), (74), (66).
(200) follows from (77), (77), (74), (74), (74), (74), (65).
(201) follows from (77), (77), (73), (73), (70), (70).
(202) follows from (77), (77), (74), (74), (71), (71).
(203) follows from (77), (77), (77), (74), (74), (74), (74), (64).
(204) follows from (77), (77), (77), (74), (74), (74), (74), (66).
(205) follows from (77), (77), (74), (74), (70), (70).
(206) follows from (74), (74), (74), (74), (72), (72), (64).
(207) follows from (74), (74), (74), (74), (72), (72), (66).
(208) follows from (77), (77), (77), (74), (74), (74), (74), (65).
(209) follows from (78), (77), (77), (73), (73), (73), (73).
(210) follows from (78), (77), (77), (74), (74), (74), (74).
(211) follows from (78), (76), (76), (71), (71), (70), (70).
(212) follows from (77), (74), (74), (74), (74), (72), (72).
(213) follows from (74), (74), (74), (74), (72), (72), (65).
(214) follows from (78), (78), (77), (73), (73), (73), (73).
(215) follows from (78), (77), (77), (74), (74), (73), (73).
(216) follows from (78), (78), (77), (74), (74), (74), (74).
(217) follows from (78), (76), (76), (71), (71), (71), (71).
(218) follows from (78), (78), (77), (74), (74), (73), (73).
(219) follows from (78), (77), (77), (77), (73), (73), (73), (73).
(220) follows from (78), (77), (77), (77), (74), (74), (74).
(221) follows from (79), (78), (74), (74), (71), (71), (70), (70).
(222) follows from (77), (77), (74), (74), (74), (74), (72), (72).
(223) follows from (78), (78), (77), (77), (73), (73), (73), (73).
(224) follows from (78), (77), (77), (77), (74), (74), (73), (73).
(225) follows from (78), (78), (77), (77), (74), (74), (74), (74).
(226) follows from (79), (78), (74), (74), (71), (71), (71), (71).
(227) follows from (78), (78), (77), (77), (74), (74), (73), (73).
(228) follows from (77), (71), (71), (66).
(229) follows from (77), (72), (72), (65).
(230) follows from (77), (72), (72), (64).
(231) follows from (78), (77), (74), (74), (66).
(232) follows from (78), (78), (74), (74), (66).
(233) follows from (77), (70), (70), (64).
(234) follows from (78), (77), (77), (74), (74), (66).
(235) follows from (77), (71), (71), (65).
(236) follows from (77), (71), (71), (64).
(237) follows from (77), (74), (74), (72), (72), (66).
(238) follows from (78), (78), (77), (74), (74), (66).
(239) follows from (78), (77), (73), (73), (64).
(240) follows from (78), (77), (74), (74), (65).
(241) follows from (78), (78), (73), (73), (64).
(242) follows from (78), (77), (74), (74), (64).
(243) follows from (78), (78), (74), (74), (65).
(244) follows from (77), (74), (74), (71), (71), (64).
(245) follows from (77), (74), (74), (71), (71), (66).
(246) follows from (78), (78), (74), (74), (64).
(247) follows from (78), (77), (77), (73), (73), (64).
(248) follows from (78), (77), (77), (74), (74), (65).
(249) follows from (77), (73), (73), (72), (72), (64).
(250) follows from (77), (74), (74), (71), (71), (65).
(251) follows from (78), (78), (77), (73), (73), (64).
(252) follows from (78), (77), (77), (74), (74), (64).
(253) follows from (77), (74), (74), (72), (72), (65).
(254) follows from (78), (78), (77), (74), (74), (65).
(255) follows from (77), (74), (74), (72), (72), (64).
(256) follows from (78), (78), (77), (74), (74), (64).
(257) follows from (78), (77), (74), (74), (74), (74), (64).
(258) follows from (78), (77), (74), (74), (74), (74), (66).
(259) follows from (78), (78), (74), (74), (74), (74), (64).
(260) follows from (78), (77), (74), (74), (74), (74), (65).
(261) follows from (78), (78), (74), (74), (74), (74), (66).
(262) follows from (78), (78), (74), (74), (74), (74), (65).
(263) follows from (78), (77), (77), (74), (74), (74), (74), (64).
(264) follows from (78), (77), (77), (74), (74), (74), (74), (66).
(265) follows from (77), (74), (74), (74), (74), (72), (72), (64).
(266) follows from (77), (74), (74), (74), (74), (72), (72), (66).
(267) follows from (78), (78), (77), (74), (74), (74), (74), (64).
(268) follows from (78), (77), (77), (74), (74), (74), (74), (65).
(269) follows from (78), (78), (77), (74), (74), (74), (74), (66).
(270) follows from (77), (74), (74), (74), (74), (72), (72), (65).
(271) follows from (78), (78), (77), (74), (74), (74), (74), (65).
(272) follows from (79).
(273) follows from (79), (77), (77).
(274) follows from (79), (78), (77).
(275) follows from (79), (78), (78).
(276) follows from (76), (76).
(277) follows from (75), (75).
(278) follows from (79), (74), (74).
(279) follows from (79), (73), (73).
(280) follows from (79), (72), (72).
(281) follows from (79), (77), (74), (74).
(282) follows from (79), (77), (73), (73).
(283) follows from (79), (77), (77), (74), (74).
(284) follows from (79), (77), (77), (73), (73).
(285) follows from (76), (76), (72), (72).
(286) follows from (75), (75), (72), (72).
(287) follows from (75), (75), (70), (70).
(288) follows from (76), (76), (71), (71).
(289) follows from (76), (76), (70), (70).
(290) follows from (79), (77), (73), (73), (73), (73).
(291) follows from (79), (77), (74), (74), (74), (74).
(292) follows from (79), (77), (74), (74), (73), (73).
(293) follows from (79), (77), (77), (73), (73), (73), (73).
(294) follows from (79), (77), (77), (74), (74), (74), (74).
(295) follows from (75), (75), (73), (73), (72), (72).
(296) follows from (76), (76), (74), (74), (72), (72).
(297) follows from (79), (77), (77), (74), (74), (73), (73).
(298) follows from (76), (76), (73), (73), (72), (72).
(299) follows from (79), (71), (71).
(300) follows from (79), (70), (70).
(301) follows from (79), (78), (74), (74).
(302) follows from (79), (78), (73), (73).
(303) follows from (78), (78), (76), (76).
(304) follows from (78), (78), (75), (75).
(305) follows from (79), (78), (77), (74), (74).
(306) follows from (79), (78), (77), (73), (73).
(307) follows from (79), (78), (78), (74), (74).
(308) follows from (79), (78), (78), (73), (73).
(309) follows from (79), (74), (74), (72), (72).
(310) follows from (79), (73), (73), (72), (72).
(311) follows from (79), (73), (73), (70), (70).
(312) follows from (79), (74), (74), (71), (71).
(313) follows from (79), (74), (74), (70), (70).
(314) follows from (79), (78), (73), (73), (73), (73).
(315) follows from (79), (78), (74), (74), (74), (74).
(316) follows from (78), (78), (75), (75), (73), (73).
(317) follows from (79), (78), (74), (74), (73), (73).
(318) follows from (78), (78), (76), (76), (74), (74).
(319) follows from (78), (78), (76), (76), (73), (73).
(320) follows from (79), (78), (77), (73), (73), (73), (73).
(321) follows from (79), (78), (77), (74), (74), (74), (74).
(322) follows from (79), (73), (73), (73), (73), (72), (72).
(323) follows from (79), (74), (74), (74), (74), (72), (72).
(324) follows from (79), (78), (78), (73), (73), (73), (73).
(325) follows from (79), (78), (77), (74), (74), (73), (73).
(326) follows from (79), (78), (78), (74), (74), (74), (74).
(327) follows from (79), (74), (74), (73), (73), (72), (72).
(328) follows from (79), (78), (78), (74), (74), (73), (73).
Jupyter Notebook for Symbolic Computations
1 Import Necessary Libraries
[ ]: import os  
from IPython.display import display  
import sympy  
from sympy import Symbol  
from utils import FileWriter, ConstraintsAggregator  
from utils import get Factors, get_term, searchFactors,→latex_repres_of_inequality
2 Initialize a File for Results
[ ]: file_path = '.../paper/result.txt'  
if os.path.exists(file_path):  
    os.remove(file_path)  
fw = FileWriter(file_path)
3 Initialize Symbols From the Paper
[ ]: rho = Symbol('rho', nonnegative=True) kappa = Symbol('kappa', nonnegative=True) lambda = Symbol('lambda', nonnegative=True) omega = Symbol('omega', nonnegative=True) p = Symbol('p', positive=True) tau = Symbol('tau', positive=True) beta = Symbol('beta', positive=True) alpha = Symbol('alpha', positive=True) n = Symbol('n', positive=True) L = Symbol('L', positive=True) L_hat = Symbol(r'\hat{ha}l\{L\}', positive=True)
L_max = Symbol(r'[L]{\max}], positive=True)  
L_bar = Symbol(r'[bar{L}'], positive=True)
4 Assistant for "First Symbolically Computed" Lemma
4.1 Calculate Expressions