5 Inequalities for the proof of the Kepler Conjecture
8 Converted from kep_inequalities.ml CVS:1.4,
9 using "modify()" in "kep_inequalities_convert.ml"
12 Eventually this file will become the final authority about
13 the various inequalities. For now, there are still typos,
14 so that 2002-version of Kepler Conjecture and the
15 interval arithmetic C++ files have higher authority.
16 The C++ code inequalities have been put into the form F < = 0.
17 Ferguson's verifications can be obtained from
18 http://www.math.pitt.edu/~thales/kepler98/samf/ferguson98.tar.gz/hales/source/
22 Acknowledgement: I would like to thank Carole Bunting for
23 typing many of these inequalities in a machine readable form.
31 Please report any errors that are found. This includes typos (such
32 as a typo in the 9-digit identifier for the inequality), missing inequalities,
33 false inequalities, incompatibilities
34 between the stated inequality and the interval arithmetic verification,
35 and incompatibilities between the stated inequality and how the inequality
36 is used in the proof of the Kepler Conjecture.
40 Nov 8, 2007: Fixed the x1 bound on calc 815492935 and
41 729988292 (SPIV-2002 Sec. A2-A3). It should be (square_2t0,x1,(#8.0))
43 Dec 16, 2007: Fixed the direction of inequalities in 690626704_*
47 (* Files for 1998 interval verification:
48 partK.cc = http://www.math.pitt.edu/~thales/kepler98/interval/partK.cc
49 533270809 appears in partK.cc but not below.
50 353116995 appears in partK.cc but not below.
51 part3.cc = http://www.math.pitt.edu/~thales/kepler98/interval/PART3/part3.c
59 (* Search for LOC: to find the location of inequalities
62 The order of the inequalities is from last paper to first:
63 Kepler Conjecture: k.c.
66 II. (a couple that are needed)
163 vor_analytic_x_flipped
181 Many of the original interval arithmetic verifications
182 were completed by Sam Ferguson. The original 1998 proof
183 (available at the arXiv)
184 contains details about which inequalities were verified by him.
192 In general, to the greatest extent possible, we express each
193 inequality as a strict inequality on a compact domain. There are,
194 however, a few inequalities that are not strict, such as the bound
195 of $1\,\pt$ on the score of a quasi_regular tetrahedron or the
196 bound of $0.0$ on the score of a quad cluster. (These particular
197 sharp bounds appear in the proof of the local optimality of the
198 face_centered cubic and hexagonal close packings.)
202 The most significant are the bounds
203 $\sigma\le\pt$ on quasi_regular tetrahedra and $\sigma\le0$ on
204 quad_clusters. The fact that these are attained for the regular cases
205 with edge lengths(#2.0) and diagonal $2\sqrt{2.0}$ on the quad_cluster
207 no other cases gives the bound $\pi/\sqrt{18.0}$ on density and the local
208 optimality of the fcc and hcp packings.
212 Another place where we have allowed equality to be obtained is
213 with $\tau_0\ge0$ for quasi_regular simplices.
217 There are also a few less significant cases where an inequality is
219 $$\tau_0(2t_0,2,2,x,2,2)\ge0,\quad\vor_0(2t_0,2,2,x,2,2)\le0$$
220 for special simplices satisfying $x\in[2\sqrt{2.0},3.2]$. Also, equality
221 occurs in Lemma~\ref{lemma:pass_makes_quarter} and
222 Lemma~\ref{lemma:neg_orient_quad}.
226 Equality is attained in \calc{} iff $S$ is a regular_tetrahedron
227 of edge_length $2.0$. Equality is attained in \calc{346093004},
228 \calc{40003553}, and \calc{522528841} \calc{892806084} iff the
229 simplex has five edges of length $2.0$ and one edge of length
234 Search for SKIP to find sections skipped.
235 Search for LOC: to find preprint locations.
238 (* avoid Jordan/parse_ext_override_interface.ml *)
240 (* real number operations *)
241 parse_as_infix("+.",(16,"right"));
242 parse_as_infix("-.",(18,"left"));
243 parse_as_infix("*.",(20,"right"));
244 parse_as_infix("**.",(24,"left"));
245 parse_as_infix("<.",(12,"right"));
246 parse_as_infix("<=.",(12,"right"));
247 parse_as_infix(">.",(12,"right"));
248 parse_as_infix(">=.",(12,"right"));
249 override_interface("+.",`real_add:real->real->real`);
250 override_interface("-.",`real_sub:real->real->real`);
251 override_interface("*.",`real_mul:real->real->real`);
252 override_interface("**.",`real_pow:real->num->real`);
254 override_interface("<.",`real_lt:real->real->bool`);
255 override_interface("<=.",`real_le:real->real->bool`);
256 override_interface(">.",`real_gt:real->real->bool`);
257 override_interface(">=.",`real_ge:real->real->bool`);
259 override_interface("--.",`real_neg:real->real`);
260 override_interface("&.",`real_of_num:num->real`);
261 override_interface("||.",`real_abs:real->real`);;
264 (* XXX Note: please don't write comments in HOL Light terms.
265 * this does not work. *)
268 LOC: 2002 k.c page 42.
272 (* interval verification in partK.cc *)
273 (* moved 572068135 to inequality_spec.ml *)
279 (* interval verification in partK.cc *)
280 (* moved 723700608 to inequality_spec.ml *)
286 (* interval verification in partK.cc *)
287 (* moved 560470084 to inequality_spec.ml *)
292 (* interval verification in partK.cc *)
293 (* moved 535502975 to inequality_spec.ml *)
300 LOC: 2002 k.c page 42
306 (* let I_821_707685= *)
307 (* all_forall `ineq *)
308 (* [((#4.0), x1, (#6.3001)); *)
309 (* ((#4.0), x2, (square (#2.168))); *)
310 (* ((#4.0), x3, (square (#2.168))); *)
311 (* ((#4.0), x4, (#6.3001)); *)
312 (* ((#4.0), x5, (#6.3001)); *)
313 (* (square_2t0, x6, square_4t0) *)
315 (* ( (dih_x x1 x2 x3 x4 x5 x6) <. (#1.63))`;; *)
317 (* Added delta_x > 0, Jan 2008 *)
318 (* interval verification by Ferguson *)
319 (* moved 821707685 to inequality_spec.ml *)
322 (* interval verification by Ferguson *)
323 (* moved 115383627 to inequality_spec.ml *)
326 (* interval verification by Ferguson *)
327 (* moved 576221766 to inequality_spec.ml *)
331 (* interval verification by Ferguson *)
332 (* moved 122081309 to inequality_spec.ml *)
336 (* interval verification by Ferguson *)
337 (* moved 644534985 to inequality_spec.ml *)
341 (* interval verification by Ferguson *)
342 (* moved 467530297 to inequality_spec.ml *)
346 (* interval verification by Ferguson *)
347 (* moved 603910880 to inequality_spec.ml *)
350 (* interval verification by Ferguson *)
351 (* moved 135427691 to inequality_spec.ml *)
354 (* interval verification by Ferguson *)
355 (* moved 60314528 to inequality_spec.ml *)
358 (* interval verification by Ferguson *)
359 (* moved 312132053 to inequality_spec.ml *)
367 LOC: 2002 k.c page 42
371 (* moved 751442360 to inequality_spec.ml *)
379 [((#4.0), x1, square_2t0);
380 ((#4.0), x2, (square (#2.168)));
381 ((#4.0), x3, (square (#2.168)));
382 ((#4.0), x4, square_2t0);
383 (square_2t0, x5, (square (#3.488)));
384 ((#4.0), x6, square_2t0)
388 ( ((tau_0_x x1 x2 x3 x4 x5 x6) ) -. ( (#0.2529) *. (dih_x x1 x2 x3 x4 x5 x6))) >.
390 ( (delta_x x5 (#4.0) (#4.0) (#8.0) square_2t0 x6) <. (#0.0)))`;;
395 Added delta constraint, 3/9/08
398 (* mistyped as 69064028 *)
399 (* moved 69064028 to inequality_spec.ml *)
406 LOC: 2002 k.c page 42
411 (* interval verification in partK.cc *)
414 [(square_2t0, x1, (#8.0));
415 ((#4.0), x2, square_2t0);
416 ((#4.0), x3, square_2t0);
417 ((square (#3.2)), x4, (square (#3.2)));
418 ((#4.0), x5, square_2t0);
419 ((#4.0), x6, square_2t0)
422 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.78)) \/
423 ( ( (sqrt x2) +. (sqrt x3)) >. (#4.6)))`;;
430 LOC: 2002 k.c page 42-43
436 (* interval verification in partK.cc *)
439 [((#4.0), x1, square_2t0);
440 ((#4.0), x2, square_2t0);
441 ((#4.0), x3, square_2t0);
442 (square_2t0, x4, (#8.0));
443 ((#4.0), x5, square_2t0);
444 ((#4.0), x6, square_2t0)
447 ( (( --. ) (dih2_x x1 x2 x3 x4 x5 x6)) +. ( (#0.35) *. (sqrt x2)) +. ( (--. (#0.15)) *.
448 (sqrt x1)) +. ( (--. (#0.15)) *. (sqrt x3)) +. ( (#0.7022) *. (sqrt x5)) +. ( (--. (#0.17)) *.
449 (sqrt x4))) >. (--. (#0.0123)))`;;
457 [((#4.0), x1, square_2t0);
458 ((#4.0), x2, square_2t0);
459 ((#4.0), x3, square_2t0);
460 (square_2t0, x4, (#8.0));
461 ((#4.0), x5, square_2t0);
462 ((#4.0), x6, square_2t0)
465 ( (dih2_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.13)) *. (sqrt x2)) +. ( (#0.631) *.
466 (sqrt x1)) +. ( (#0.31) *. (sqrt x3)) +. ( (--. (#0.58)) *. (sqrt x5)) +. ( (#0.413) *.
467 (sqrt x4)) +. ( (#0.025) *. (sqrt x6))) >. (#2.63363))`;;
473 [((#4.0), x1, square_2t0);
474 ((#4.0), x2, square_2t0);
475 ((#4.0), x3, square_2t0);
476 (square_2t0, x4, (#8.0));
477 ((#4.0), x5, square_2t0);
478 ((#4.0), x6, square_2t0)
481 ( (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) +. ( (#0.714) *. (sqrt x1)) +. ( (--. (#0.221)) *.
482 (sqrt x2)) +. ( (--. (#0.221)) *. (sqrt x3)) +. ( (#0.92) *. (sqrt x4)) +. ( (--. (#0.221)) *.
483 (sqrt x5)) +. ( (--. (#0.221)) *. (sqrt x6))) >. (#0.3482))`;;
490 [((#4.0), x1, square_2t0);
491 ((#4.0), x2, square_2t0);
492 ((#4.0), x3, square_2t0);
493 (square_2t0, x4, (#8.0));
494 ((#4.0), x5, square_2t0);
495 ((#4.0), x6, square_2t0)
498 ( (dih_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.315)) *. (sqrt x1)) +. ( (#0.3972) *.
499 (sqrt x2)) +. ( (#0.3972) *. (sqrt x3)) +. ( (--. (#0.715)) *. (sqrt x4)) +. ( (#0.3972) *.
500 (sqrt x5)) +. ( (#0.3972) *. (sqrt x6))) >. (#2.37095))`;;
503 (* interval verification by Ferguson *)
506 [((#4.0), x1, square_2t0);
507 ((#4.0), x2, square_2t0);
508 ((#4.0), x3, square_2t0);
509 (square_2t0, x4, (#8.0));
510 ((#4.0), x5, square_2t0);
511 ((#4.0), x6, square_2t0)
514 ( (( --. ) (sol_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.187)) *. (sqrt x1)) +. ( (--. (#0.187)) *.
515 (sqrt x2)) +. ( (--. (#0.187)) *. (sqrt x3)) +. ( (#0.1185) *. (sqrt x4)) +. ( (#0.479) *.
516 (sqrt x5)) +. ( (#0.479) *. (sqrt x6))) >. (#0.437235))`;;
519 (* interval verification by Ferguson *)
522 [((#4.0), x1, square_2t0);
523 ((#4.0), x2, square_2t0);
524 ((#4.0), x3, square_2t0);
525 (square_2t0, x4, (#8.0));
526 ((#4.0), x5, square_2t0);
527 ((#4.0), x6, square_2t0)
530 ( (sol_x x1 x2 x3 x4 x5 x6) +. ( (#0.488) *. (sqrt x1)) +. ( (#0.488) *.
531 (sqrt x2)) +. ( (#0.488) *. (sqrt x3)) +. ( (--. (#0.334)) *. (sqrt x5)) +. ( (--. (#0.334)) *.
532 (sqrt x6))) >. (#2.244))`;;
538 [((#4.0), x1, square_2t0);
539 ((#4.0), x2, square_2t0);
540 ((#4.0), x3, square_2t0);
541 (square_2t0, x4, (#8.0));
542 ((#4.0), x5, square_2t0);
543 ((#4.0), x6, square_2t0)
546 ( (( --. ) (sigmahat_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.145)) *. (sqrt x1)) +. ( (--. (#0.081)) *.
547 (sqrt x2)) +. ( (--. (#0.081)) *. (sqrt x3)) +. ( (--. (#0.133)) *. (sqrt x5)) +. ( (--. (#0.133)) *.
548 (sqrt x6))) >. (--. (#1.17401)))`;;
552 [((#4.0), x1, square_2t0);
553 ((#4.0), x2, square_2t0);
554 ((#4.0), x3, square_2t0);
555 (square_2t0, x4, (#8.0));
556 ((#4.0), x5, square_2t0);
557 ((#4.0), x6, square_2t0)
560 ( (( --. ) (sigmahat_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.12)) *. (sqrt x1)) +. ( (--. (#0.081)) *.
561 (sqrt x2)) +. ( (--. (#0.081)) *. (sqrt x3)) +. ( (--. (#0.113)) *. (sqrt x5)) +. ( (--. (#0.113)) *.
562 (sqrt x6)) +. ( (#0.029) *. (sqrt x4))) >. (--. (#0.94903)))`;;
568 [((#4.0), x1, square_2t0);
569 ((#4.0), x2, square_2t0);
570 ((#4.0), x3, square_2t0);
571 (square_2t0, x4, (#8.0));
572 ((#4.0), x5, square_2t0);
573 ((#4.0), x6, square_2t0)
576 ( (sigmahat_x x1 x2 x3 x4 x5 x6) +. ( (#0.153) *. (sqrt x4)) +. ( (#0.153) *.
577 (sqrt x5)) +. ( (#0.153) *. (sqrt x6))) <. (#1.05382))`;;
581 [((#4.0), x1, square_2t0);
582 ((#4.0), x2, square_2t0);
583 ((#4.0), x3, square_2t0);
584 (square_2t0, x4, (#8.0));
585 ((#4.0), x5, square_2t0);
586 ((#4.0), x6, square_2t0)
589 ( (sigmahat_x x1 x2 x3 x4 x5 x6) +. ( (#0.419351) *. (sol_x x1 x2 x3 x4 x5 x6)) +. ( (#0.19) *.
590 (sqrt x1)) +. ( (#0.19) *. (sqrt x2)) +. ( (#0.19) *. (sqrt x3))) <. (#1.449))`;;
595 [((#4.0), x1, square_2t0);
596 ((#4.0), x2, square_2t0);
597 ((#4.0), x3, square_2t0);
598 (square_2t0, x4, (#8.0));
599 ((#4.0), x5, square_2t0);
600 ((#4.0), x6, square_2t0)
603 ( (sigmahat_x x1 x2 x3 x4 x5 x6) +. ( (#0.419351) *. (sol_x x1 x2 x3 x4 x5 x6))) <.
604 ( (--. (#0.01465)) +. ( (#0.0436) *. (sqrt x5)) +. ( (#0.436) *. (sqrt x6)) +. ( (#0.079431) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
609 [((#4.0), x1, square_2t0);
610 ((#4.0), x2, square_2t0);
611 ((#4.0), x3, square_2t0);
612 (square_2t0, x4, (#8.0));
613 ((#4.0), x5, square_2t0);
614 ((#4.0), x6, square_2t0)
616 ( (sigmahat_x x1 x2 x3 x4 x5 x6) <. (#0.0114))`;;
621 [((#4.0), x1, square_2t0);
622 ((#4.0), x2, square_2t0);
623 ((#4.0), x3, square_2t0);
624 (square_2t0, x4, (#8.0));
625 ((#4.0), x5, square_2t0);
626 ((#4.0), x6, square_2t0)
628 ( (tauhat_x x1 x2 x3 x4 x5 x6) >. ( (#1.019) *. pt))`;;
633 LOC: 2002 k.c page 43
638 (* let I_498839271_1= *)
639 (* all_forall `ineq *)
640 (* [(square_2t0, x1, (#8.0)); *)
641 (* ((#4.0), x2, square_2t0); *)
642 (* ((#4.0), x3, square_2t0); *)
643 (* ((#4.0), x4, square_2t0); *)
644 (* ((#4.0), x5, square_2t0); *)
645 (* ((#4.0), x6, square_2t0) *)
647 (* ( (sqrt x1) >. (#2.51))`;; *)
652 (* let I_498839271_2= *)
653 (* all_forall `ineq *)
654 (* [(square_2t0, x1, (#8.0)); *)
655 (* ((#4.0), x2, square_2t0); *)
656 (* ((#4.0), x3, square_2t0); *)
657 (* ((#4.0), x4, square_2t0); *)
658 (* ((#4.0), x5, square_2t0); *)
659 (* ((#4.0), x6, square_2t0) *)
661 (* ( (sqrt x1) <=. ( (#2.0) *. (sqrt (#2.0))))`;; *)
665 (* interval verification in partK.cc *)
667 (* CCC Shouldn't this say > rather than >= ?
673 [(square_2t0, x1, (#8.0));
674 ((#4.0), x2, square_2t0);
675 ((#4.0), x3, square_2t0);
676 ((#4.0), x4, square_2t0);
677 ((#4.0), x5, square_2t0);
678 ((#4.0), x6, square_2t0)
681 ( (dih_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.636)) *. (sqrt x1)) +. ( (#0.462) *. (sqrt x2)) +. ( (#0.462) *. (sqrt x3)) +.
682 ( (--. (#0.82)) *. (sqrt x4)) +. ( (#0.462) *. (sqrt x5)) +. ( (#0.462) *. (sqrt x6))) >. (#1.82419))`;;
685 (* interval verification in partK.cc *)
688 [(square_2t0, x1, (#8.0));
689 ((#4.0), x2, square_2t0);
690 ((#4.0), x3, square_2t0);
691 ((#4.0), x4, square_2t0);
692 ((#4.0), x5, square_2t0);
693 ((#4.0), x6, square_2t0)
696 ( (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) +. ( (#0.55) *. (sqrt x1)) +. ( (--. (#0.214)) *. (sqrt x2)) +. ( (--. (#0.214)) *. (sqrt x3)) +.
697 ( (#1.24) *. (sqrt x4)) +. ( (--. (#0.214)) *. (sqrt x5)) +. ( (--. (#0.214)) *. (sqrt x6))) >. (#0.75281))`;;
699 (* interval verification in partK.cc *)
702 [(square_2t0, x1, (#8.0));
703 ((#4.0), x2, square_2t0);
704 ((#4.0), x3, square_2t0);
705 ((#4.0), x4, square_2t0);
706 ((#4.0), x5, square_2t0);
707 ((#4.0), x6, square_2t0)
710 ( (dih2_x x1 x2 x3 x4 x5 x6) +. ( (#0.4) *. (sqrt x1)) +. ( (--. (#0.15)) *. (sqrt x2)) +. ( (#0.09) *. (sqrt x3)) +.
711 ( (#0.631) *. (sqrt x4)) +. ( (--. (#0.57)) *. (sqrt x5)) +. ( (#0.23) *. (sqrt x6))) >. (#2.5481))`;;
714 (* interval verification in partK.cc *)
717 [(square_2t0, x1, (#8.0));
718 ((#4.0), x2, square_2t0);
719 ((#4.0), x3, square_2t0);
720 ((#4.0), x4, square_2t0);
721 ((#4.0), x5, square_2t0);
722 ((#4.0), x6, square_2t0)
725 ( (( --. ) (dih2_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.454)) *. (sqrt x1)) +. ( (#0.34) *. (sqrt x2)) +. ( (#1.54) *. (sqrt x3)) +.
726 ( (--. (#0.346)) *. (sqrt x4)) +. ( (#0.805) *. (sqrt x5))) >. (--. (#0.3429)))`;;
729 (* interval verification in partK.cc *)
732 [(square_2t0, x1, (#8.0));
733 ((#4.0), x2, square_2t0);
734 ((#4.0), x3, square_2t0);
735 ((#4.0), x4, square_2t0);
736 ((#4.0), x5, square_2t0);
737 ((#4.0), x6, square_2t0)
740 ( (dih3_x x1 x2 x3 x4 x5 x6) +. ( (#0.4) *. (sqrt x1)) +. ( (--. (#0.15)) *. (sqrt x3)) +. ( (#0.09) *. (sqrt x2)) +.
741 ( (#0.631) *. (sqrt x4)) +. ( (--. (#0.57)) *. (sqrt x6)) +. ( (#0.23) *. (sqrt x5))) >. (#2.5481))`;;
746 (* Seems to be wrong : check at
747 (8, 4.77946715116, 4.0, 6.30009999999, 6.30009999999, 4)
748 STM changed from 0.364
749 1/20/2008. This seems to fix the problem. The
750 left hand side evaluates to -0.342688 > -0.3429.
752 (* interval verification in partK.cc *)
755 [(square_2t0, x1, (#8.0));
756 ((#4.0), x2, square_2t0);
757 ((#4.0), x3, square_2t0);
758 ((#4.0), x4, square_2t0);
759 ((#4.0), x5, square_2t0);
760 ((#4.0), x6, square_2t0)
763 ( (( --. ) (dih3_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.454)) *. (sqrt x1)) +. ( (#0.34) *. (sqrt x3)) +. ( (#0.154) *. (sqrt x2)) +.
764 ( (--. (#0.346)) *. (sqrt x4)) +. ( (#0.805) *. (sqrt x6))) >. (--. (#0.3429)))`;;
767 (* interval verification in partK.cc *)
770 [(square_2t0, x1, (#8.0));
771 ((#4.0), x2, square_2t0);
772 ((#4.0), x3, square_2t0);
773 ((#4.0), x4, square_2t0);
774 ((#4.0), x5, square_2t0);
775 ((#4.0), x6, square_2t0)
778 ( (sol_x x1 x2 x3 x4 x5 x6) +. ( (#0.065) *. (sqrt x2)) +. ( (#0.065) *. (sqrt x3)) +. ( (#0.061) *. (sqrt x4)) +.
779 ( (--. (#0.115)) *. (sqrt x5)) +. ( (--. (#0.115)) *. (sqrt x6))) >. (#0.2618))`;;
782 (* interval verification in partK.cc *)
785 [(square_2t0, x1, (#8.0));
786 ((#4.0), x2, square_2t0);
787 ((#4.0), x3, square_2t0);
788 ((#4.0), x4, square_2t0);
789 ((#4.0), x5, square_2t0);
790 ((#4.0), x6, square_2t0)
793 ( (( --. ) (sol_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.293)) *. (sqrt x1)) +. ( (--. (#0.03)) *. (sqrt x2)) +. ( (--. (#0.03)) *. (sqrt x3)) +.
794 ( (#0.12) *. (sqrt x4)) +. ( (#0.325) *. (sqrt x5)) +. ( (#0.325) *. (sqrt x6))) >. (#0.2514))`;;
797 (* interval verification in partK.cc *)
800 [(square_2t0, x1, (#8.0));
801 ((#4.0), x2, square_2t0);
802 ((#4.0), x3, square_2t0);
803 ((#4.0), x4, square_2t0);
804 ((#4.0), x5, square_2t0);
805 ((#4.0), x6, square_2t0)
808 ( (( --. ) (nu_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.0538)) *. (sqrt x2)) +. ( (--. (#0.0538)) *. (sqrt x3)) +.
809 ( (--. (#0.083)) *. (sqrt x4)) +. ( (--. (#0.0538)) *. (sqrt x5)) +. ( (--. (#0.0538)) *. (sqrt x6))) >. (--. (#0.5995)))`;;
811 (* interval verification in partK.cc *)
814 [(square_2t0, x1, (#8.0));
815 ((#4.0), x2, square_2t0);
816 ((#4.0), x3, square_2t0);
817 ((#4.0), x4, square_2t0);
818 ((#4.0), x5, square_2t0);
819 ((#4.0), x6, square_2t0)
821 ( (nu_x x1 x2 x3 x4 x5 x6) >=. (#0.0))`;;
824 (* interval verification in partK.cc *)
827 [(square_2t0, x1, (#8.0));
828 ((#4.0), x2, square_2t0);
829 ((#4.0), x3, square_2t0);
830 ((#4.0), x4, square_2t0);
831 ((#4.0), x5, square_2t0);
832 ((#4.0), x6, square_2t0)
835 ( (taunu_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.5945)) *. pt)) >. (#0.0))`;;
842 LOC: 2002 k.c page 45
847 (* interval verification in partK.cc *)
850 [(square_2t0, x1, (square (#2.696)));
851 ((#4.0), x2, square_2t0);
852 ((#4.0), x3, square_2t0);
853 ((#4.0), x4, square_2t0);
854 ((#4.0), x5, square_2t0);
855 ((#4.0), x6, square_2t0)
857 ( (sqrt x1) <. (#2.696))`;;
864 [(square_2t0, x1, (square (#2.696)));
865 ((#4.0), x2, square_2t0);
866 ((#4.0), x3, square_2t0);
867 ((#4.0), x4, square_2t0);
868 ((#4.0), x5, square_2t0);
869 ((#4.0), x6, square_2t0)
872 ( (dih_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.49)) *. (sqrt x1)) +. ( (#0.44) *. (sqrt x2)) +. ( (#0.44) *. (sqrt x3)) +.
873 ( (--. (#0.82)) *. (sqrt x4)) +. ( (#0.44) *. (sqrt x5)) +. ( (#0.44) *. (sqrt x6))) >. (#2.0421))`;;
879 [(square_2t0, x1, (square (#2.696)));
880 ((#4.0), x2, square_2t0);
881 ((#4.0), x3, square_2t0);
882 ((#4.0), x4, square_2t0);
883 ((#4.0), x5, square_2t0);
884 ((#4.0), x6, square_2t0)
887 ( (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) +. ( (#0.495) *. (sqrt x1)) +. ( (--. (#0.214)) *. (sqrt x2)) +. ( (--. (#0.214)) *. (sqrt x3)) +.
888 ( (#1.05) *. (sqrt x4)) +. ( (--. (#0.214)) *. (sqrt x5)) +. ( (--. (#0.214)) *. (sqrt x6))) >. (#0.2282))`;;
894 [(square_2t0, x1, (square (#2.696)));
895 ((#4.0), x2, square_2t0);
896 ((#4.0), x3, square_2t0);
897 ((#4.0), x4, square_2t0);
898 ((#4.0), x5, square_2t0);
899 ((#4.0), x6, square_2t0)
902 ( (dih2_x x1 x2 x3 x4 x5 x6) +. ( (#0.38) *. (sqrt x1)) +. ( (--. (#0.15)) *. (sqrt x2)) +. ( (#0.09) *. (sqrt x3)) +.
903 ( (#0.54) *. (sqrt x4)) +. ( (--. (#0.57)) *. (sqrt x5)) +. ( (#0.24) *. (sqrt x6))) >. (#2.3398))`;;
909 [(square_2t0, x1, (square (#2.696)));
910 ((#4.0), x2, square_2t0);
911 ((#4.0), x3, square_2t0);
912 ((#4.0), x4, square_2t0);
913 ((#4.0), x5, square_2t0);
914 ((#4.0), x6, square_2t0)
917 ( (( --. ) (dih2_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.375)) *. (sqrt x1)) +. ( (#0.33) *. (sqrt x2)) +. ( (#0.11) *. (sqrt x3)) +.
918 ( (--. (#0.36)) *. (sqrt x4)) +. ( (#0.72) *. (sqrt x5)) +. ( (#0.034) *. (sqrt x6))) >. (--. (#0.36135)))`;;
923 [(square_2t0, x1, (square (#2.696)));
924 ((#4.0), x2, square_2t0);
925 ((#4.0), x3, square_2t0);
926 ((#4.0), x4, square_2t0);
927 ((#4.0), x5, square_2t0);
928 ((#4.0), x6, square_2t0)
931 ( (sol_x x1 x2 x3 x4 x5 x6) +. ( (#0.42) *. (sqrt x1)) +. ( (#0.165) *. (sqrt x2)) +. ( (#0.165) *. (sqrt x3)) +.
932 ( (--. (#0.06)) *. (sqrt x4)) +. ( (--. (#0.135)) *. (sqrt x5)) +. ( (--. (#0.135)) *. (sqrt x6))) >. (#1.479))`;;
937 [(square_2t0, x1, (square (#2.696)));
938 ((#4.0), x2, square_2t0);
939 ((#4.0), x3, square_2t0);
940 ((#4.0), x4, square_2t0);
941 ((#4.0), x5, square_2t0);
942 ((#4.0), x6, square_2t0)
945 ( (( --. ) (sol_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.265)) *. (sqrt x1)) +. ( (--. (#0.06)) *. (sqrt x2)) +. ( (--. (#0.06)) *. (sqrt x3)) +.
946 ( (#0.124) *. (sqrt x4)) +. ( (#0.296) *. (sqrt x5)) +. ( (#0.296) *. (sqrt x6))) >. (#0.0997))`;;
952 [(square_2t0, x1, (square (#2.696)));
953 ((#4.0), x2, square_2t0);
954 ((#4.0), x3, square_2t0);
955 ((#4.0), x4, square_2t0);
956 ((#4.0), x5, square_2t0);
957 ((#4.0), x6, square_2t0)
960 ( (( --. ) (nu_x x1 x2 x3 x4 x5 x6)) +. ( (#0.112) *. (sqrt x1)) +. ( (--. (#0.142)) *. (sqrt x2)) +. ( (--. (#0.142)) *. (sqrt x3)) +.
961 ( (--. (#0.16)) *. (sqrt x4)) +. ( (--. (#0.074)) *. (sqrt x5)) +. ( (--. (#0.074)) *. (sqrt x6))) >. (--. (#0.9029)))`;;
967 [(square_2t0, x1, (square (#2.696)));
968 ((#4.0), x2, square_2t0);
969 ((#4.0), x3, square_2t0);
970 ((#4.0), x4, square_2t0);
971 ((#4.0), x5, square_2t0);
972 ((#4.0), x6, square_2t0)
975 ( (nu_x x1 x2 x3 x4 x5 x6) +. ( (#0.07611) *. (dih_x x1 x2 x3 x4 x5 x6))) <. (#0.11))`;;
981 Bound: 0.855729929143
982 Point: [6.30009999999, 5.76256763219, 6.30009999999, 6.30009999999, 6.30009999999, 5.92418597238]
984 There is a sign error in the statement of the inequality
985 in SPVI2002:page44. It should be -nu_gamma_x.
986 A note has been added to the dcg_errata (even though it is not an error there).
988 The interval arithmetic file partK.c (1998) states it correctly.
992 [(square_2t0, x1, (square (#2.696)));
993 ((#4.0), x2, square_2t0);
994 ((#4.0), x3, square_2t0);
995 ((#4.0), x4, square_2t0);
996 ((#4.0), x5, square_2t0);
997 ((#4.0), x6, square_2t0)
1000 ((--. (nu_gamma_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.015)) *. (sqrt x1)) +. ( (--. (#0.16)) *. ( (sqrt x2) +. (sqrt x3) +. (sqrt x4))) +.
1001 ( (--. (#0.0738)) *. ( (sqrt x5) +. (sqrt x6))) ) >. (--. (#1.29285)))
1002 \/ (sqrt2 <. (eta_x x1 x2 x6) ))`;;
1009 [(square_2t0, x1, (square (#2.696)));
1010 ((#4.0), x2, square_2t0);
1011 ((#4.0), x3, square_2t0);
1012 ((#4.0), x4, square_2t0);
1013 ((#4.0), x5, square_2t0);
1014 ((#4.0), x6, square_2t0)
1017 ( (taunu_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.07106)) *. (dih_x x1 x2 x3 x4 x5 x6))) >. (--. (#0.06429)))`;;
1024 [(square_2t0, x1, (square (#2.696)));
1025 ((#4.0), x2, square_2t0);
1026 ((#4.0), x3, square_2t0);
1027 ((#4.0), x4, square_2t0);
1028 ((#4.0), x5, square_2t0);
1029 ((#4.0), x6, square_2t0)
1031 ( (taunu_x x1 x2 x3 x4 x5 x6) >. (#0.0414))`;;
1036 LOC: 2002 k.c page 44
1041 In connection with the Inequality (I_319046543_3), we
1042 occasionally use the stronger constant $0.2345$ instead of
1043 $0.2282$. To justify this constant, we have checked using
1044 interval arithmetic that the bound $0.2345$ holds if $y_1\le2.68$
1045 or $y_4\le2.475$. Further interval calculations show that the
1046 anchored simplices can be erased if they share an upright diagonal
1047 with such a quarter.
1054 [(square_2t0, x1, (square (#2.696)));
1055 ((#4.0), x2, square_2t0);
1056 ((#4.0), x3, square_2t0);
1057 ((#4.0), x4, square_2t0);
1058 ((#4.0), x5, square_2t0);
1059 ((#4.0), x6, square_2t0)
1063 ( (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) +. ( (#0.495) *. (sqrt x1)) +. ( (--. (#0.214)) *. (sqrt x2)) +. ( (--. (#0.214)) *. (sqrt x3)) +.
1064 ( (#1.05) *. (sqrt x4)) +. ( (--. (#0.214)) *. (sqrt x5)) +. ( (--. (#0.214)) *. (sqrt x6))) >. (#0.2345)) \/
1065 ( (sqrt x1) >. (#2.68)))`;;
1072 [(square_2t0, x1, (square (#2.696)));
1073 ((#4.0), x2, square_2t0);
1074 ((#4.0), x3, square_2t0);
1075 ((#4.0), x4, square_2t0);
1076 ((#4.0), x5, square_2t0);
1077 ((#4.0), x6, square_2t0)
1081 ( (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) +. ( (#0.495) *. (sqrt x1)) +. ( (--. (#0.214)) *. (sqrt x2)) +. ( (--. (#0.214)) *. (sqrt x3)) +.
1082 ( (#1.05) *. (sqrt x4)) +. ( (--. (#0.214)) *. (sqrt x5)) +. ( (--. (#0.214)) *. (sqrt x6))) >. (#0.2345)) \/
1083 ( (sqrt x4) >. (#2.475)))`;;
1091 LOC: 2002 k.c page 44--45
1096 The following comment about Group_8 is copied from
1097 KC_2002_17.8_page44_group8.
1101 We give lower and upper bounds on dihedral angles. The domains that we
1102 list are not disjoint. In general we consider an edge as belonging to
1103 the most restrictive domain that the information of the following charts
1104 permit us to conclude that it lies in.
1109 (* interval verification by Ferguson *)
1112 [((#4.0), x1, square_2t0);
1113 ((#4.0), x2, square_2t0);
1114 ((#4.0), x3, square_2t0);
1115 (square_2t0, x4, (#8.0));
1116 ((#4.0), x5, square_2t0);
1117 ((#4.0), x6, square_2t0)
1119 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.153))`;;
1124 (* interval verification by Ferguson *)
1127 [((#4.0), x1, square_2t0);
1128 ((#4.0), x2, square_2t0);
1129 ((#4.0), x3, square_2t0);
1130 (square_2t0, x4, (#8.0));
1131 ((#4.0), x5, square_2t0);
1132 ((#4.0), x6, square_2t0)
1134 ( (dih_x x1 x2 x3 x4 x5 x6) <. (#2.28))`;;
1140 (* interval verification by Ferguson *)
1141 (* Uses monotonicity reduction in x4 variable *)
1144 [((#4.0), x1, square_2t0);
1145 ((#4.0), x2, square_2t0);
1146 ((#4.0), x3, square_2t0);
1147 ((#8.0), x4, (#8.0));
1148 ((#4.0), x5, square_2t0);
1149 ((#4.0), x6, square_2t0)
1151 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.32))`;;
1156 (* interval verification by Ferguson *)
1157 (* By definition dih <= pi, so there is no need for intervals here *)
1162 [((#4.0), x1, square_2t0);
1163 ((#4.0), x2, square_2t0);
1164 ((#4.0), x3, square_2t0);
1165 ((#8.0), x4, square_4t0);
1166 ((#4.0), x5, square_2t0);
1167 ((#4.0), x6, square_2t0)
1169 ( (dih_x x1 x2 x3 x4 x5 x6) <. ( (#2.0) *. pi))`;;
1174 (* interval verification by Ferguson *)
1177 [((#4.0), x1, square_2t0);
1178 ((#4.0), x2, square_2t0);
1179 ((#4.0), x3, square_2t0);
1180 ((#4.0), x4, square_2t0);
1181 ((#4.0), x5, square_2t0);
1182 (square_2t0, x6, (#8.0))
1184 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#0.633))`;;
1189 (* interval verification by Ferguson *)
1192 [((#4.0), x1, square_2t0);
1193 ((#4.0), x2, square_2t0);
1194 ((#4.0), x3, square_2t0);
1195 ((#4.0), x4, square_2t0);
1196 ((#4.0), x5, square_2t0);
1197 (square_2t0, x6, (square (#3.02)))
1199 ( (dih_x x1 x2 x3 x4 x5 x6) <. (#1.624))`;;
1205 (* interval verification by Ferguson *)
1208 [((#4.0), x1, square_2t0);
1209 ((#4.0), x2, square_2t0);
1210 ((#4.0), x3, square_2t0);
1211 (square_2t0, x4, (#8.0));
1212 ((#4.0), x5, square_2t0);
1213 (square_2t0, x6, (#8.0))
1215 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.033))`;;
1220 (* interval verification by Ferguson *)
1223 [((#4.0), x1, square_2t0);
1224 ((#4.0), x2, square_2t0);
1225 ((#4.0), x3, square_2t0);
1226 (square_2t0, x4, (#8.0));
1227 ((#4.0), x5, square_2t0);
1228 (square_2t0, x6, (#8.0))
1230 ( (dih_x x1 x2 x3 x4 x5 x6) <. (#1.929))`;;
1236 (* interval verification by Ferguson *)
1239 [((#4.0), x1, square_2t0);
1240 ((#4.0), x2, square_2t0);
1241 ((#4.0), x3, square_2t0);
1242 (square_2t0, x4, square_4t0);
1243 ((#4.0), x5, square_2t0);
1244 (square_2t0, x6, (#8.0))
1246 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.033))`;;
1251 (* interval verification by Ferguson *)
1254 [((#4.0), x1, square_2t0);
1255 ((#4.0), x2, square_2t0);
1256 ((#4.0), x3, square_2t0);
1257 (square_2t0, x4, square_4t0);
1258 ((#4.0), x5, square_2t0);
1259 (square_2t0, x6, (#8.0))
1261 ( (dih_x x1 x2 x3 x4 x5 x6) <. ( (#2.0) *. pi))`;;
1266 (* interval verification by Ferguson *)
1269 [((#4.0), x1, square_2t0);
1270 ((#4.0), x2, square_2t0);
1271 ((#4.0), x3, square_2t0);
1272 ((#8.0), x4, square_4t0);
1273 ((#4.0), x5, square_2t0);
1274 (square_2t0, x6, (#8.0))
1276 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.259))`;;
1281 (* interval verification by Ferguson *)
1284 [((#4.0), x1, square_2t0);
1285 ((#4.0), x2, square_2t0);
1286 ((#4.0), x3, square_2t0);
1287 ((#8.0), x4, square_4t0);
1288 ((#4.0), x5, square_2t0);
1289 (square_2t0, x6, (#8.0))
1291 ( (dih_x x1 x2 x3 x4 x5 x6) <. ( (#2.0) *. pi))`;;
1297 (* interval verification by Ferguson *)
1300 [((#4.0), x1, square_2t0);
1301 ((#4.0), x2, square_2t0);
1302 ((#4.0), x3, square_2t0);
1303 ((#4.0), x4, square_2t0);
1304 (square_2t0, x5, (#8.0));
1305 (square_2t0, x6, (#8.0))
1307 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#0.817))`;;
1312 (* interval verification by Ferguson *)
1315 [((#4.0), x1, square_2t0);
1316 ((#4.0), x2, square_2t0);
1317 ((#4.0), x3, square_2t0);
1318 ((#4.0), x4, square_2t0);
1319 (square_2t0, x5, (square (#3.02)));
1320 (square_2t0, x6, (square (#3.02)))
1322 ( (dih_x x1 x2 x3 x4 x5 x6) <. (#1.507))`;;
1327 (* interval verification by Ferguson *)
1330 [((#4.0), x1, square_2t0);
1331 ((#4.0), x2, square_2t0);
1332 ((#4.0), x3, square_2t0);
1333 (square_2t0, x4, (#8.0));
1334 (square_2t0, x5, (#8.0));
1335 (square_2t0, x6, (#8.0))
1337 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.07))`;;
1342 (* interval verification by Ferguson *)
1345 [((#4.0), x1, square_2t0);
1346 ((#4.0), x2, square_2t0);
1347 ((#4.0), x3, square_2t0);
1348 (square_2t0, x4, (#8.0));
1349 (square_2t0, x5, (#8.0));
1350 (square_2t0, x6, (#8.0))
1352 ( (dih_x x1 x2 x3 x4 x5 x6) <. (#1.761))`;;
1357 (* interval verification by Ferguson *)
1360 [((#4.0), x1, square_2t0);
1361 ((#4.0), x2, square_2t0);
1362 ((#4.0), x3, square_2t0);
1363 (square_2t0, x4, square_4t0);
1364 (square_2t0, x5, (#8.0));
1365 (square_2t0, x6, (#8.0))
1367 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.07))`;;
1372 (* interval verification by Ferguson *)
1375 [((#4.0), x1, square_2t0);
1376 ((#4.0), x2, square_2t0);
1377 ((#4.0), x3, square_2t0);
1378 (square_2t0, x4, square_4t0);
1379 (square_2t0, x5, (#8.0));
1380 (square_2t0, x6, (#8.0))
1382 ( (dih_x x1 x2 x3 x4 x5 x6) <. ( (#2.0) *. pi))`;;
1387 (* interval verification by Ferguson *)
1390 [((#4.0), x1, square_2t0);
1391 ((#4.0), x2, square_2t0);
1392 ((#4.0), x3, square_2t0);
1393 ((#8.0), x4, square_4t0);
1394 (square_2t0, x5, (#8.0));
1395 (square_2t0, x6, (#8.0))
1397 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.23))`;;
1402 (* interval verification by Ferguson *)
1405 [((#4.0), x1, square_2t0);
1406 ((#4.0), x2, square_2t0);
1407 ((#4.0), x3, square_2t0);
1408 ((#8.0), x4, square_4t0);
1409 (square_2t0, x5, (#8.0));
1410 (square_2t0, x6, (#8.0))
1412 ( (dih_x x1 x2 x3 x4 x5 x6) <. ( (#2.0) *. pi))`;;
1417 (* interval verification by Ferguson *)
1420 [(square_2t0, x1, (#8.0));
1421 ((#4.0), x2, square_2t0);
1422 ((#4.0), x3, square_2t0);
1423 ((#4.0), x4, square_2t0);
1424 ((#4.0), x5, square_2t0);
1425 ((#4.0), x6, square_2t0)
1427 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#0.956))`;;
1431 (* interval verification by Ferguson *)
1434 [(square_2t0, x1, (#8.0));
1435 ((#4.0), x2, square_2t0);
1436 ((#4.0), x3, square_2t0);
1437 ((#4.0), x4, square_2t0);
1438 ((#4.0), x5, square_2t0);
1439 ((#4.0), x6, square_2t0)
1441 ( (dih_x x1 x2 x3 x4 x5 x6) <. (#2.184))`;;
1446 (* interval verification by Ferguson *)
1449 [(square_2t0, x1, (#8.0));
1450 ((#4.0), x2, square_2t0);
1451 ((#4.0), x3, square_2t0);
1452 (square_2t0, x4, (#8.0));
1453 ((#4.0), x5, square_2t0);
1454 ((#4.0), x6, square_2t0)
1456 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.23))`;;
1463 (* interval verification by Ferguson *)
1464 (* Uses monotonicity in the x4 variable *)
1468 [(square_2t0, x1, (#8.0));
1469 ((#4.0), x2, square_2t0);
1470 ((#4.0), x3, square_2t0);
1471 (square_2t0, x4, square_2t0);
1472 ((#4.0), x5, square_2t0);
1473 ((#4.0), x6, square_2t0)
1475 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.23))`;;
1480 (* interval verification by Ferguson *)
1481 (* Uses monotonicity in the x4 variable *)
1485 [(square_2t0, x1, (#8.0));
1486 ((#4.0), x2, square_2t0);
1487 ((#4.0), x3, square_2t0);
1488 ((#8.0), x4, (#8.0));
1489 ((#4.0), x5, square_2t0);
1490 ((#4.0), x6, square_2t0)
1492 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.416))`;;
1494 (* interval verification by Ferguson *)
1497 [((#4.0), x1, square_2t0);
1498 ((#4.0), x2, square_2t0);
1499 (square_2t0, x3, (#8.0));
1500 ((#4.0), x4, square_2t0);
1501 ((#4.0), x5, square_2t0);
1502 ((#4.0), x6, square_2t0)
1504 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#0.633))`;;
1509 (* interval verification by Ferguson *)
1512 [((#4.0), x1, square_2t0);
1513 ((#4.0), x2, square_2t0);
1514 (square_2t0, x3, (#8.0));
1515 ((#4.0), x4, square_2t0);
1516 ((#4.0), x5, square_2t0);
1517 ((#4.0), x6, square_2t0)
1519 ( (dih_x x1 x2 x3 x4 x5 x6) <. (#1.624))`;;
1524 (* interval verification by Ferguson *)
1527 [((#4.0), x1, square_2t0);
1528 ((#4.0), x2, square_2t0);
1529 (square_2t0, x3, (#8.0));
1530 (square_2t0, x4, square_2t0);
1531 ((#4.0), x5, square_2t0);
1532 ((#4.0), x6, square_2t0)
1534 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.033))`;;
1539 (* interval verification by Ferguson *)
1542 [((#4.0), x1, square_2t0);
1543 ((#4.0), x2, square_2t0);
1544 (square_2t0, x3, (#8.0));
1545 (square_2t0, x4, square_2t0);
1546 ((#4.0), x5, square_2t0);
1547 ((#4.0), x6, square_2t0)
1549 ( (dih_x x1 x2 x3 x4 x5 x6) <. ( (#2.0) *. pi))`;;
1551 (* interval verification by Ferguson *)
1554 [((#4.0), x1, square_2t0);
1555 ((#4.0), x2, square_2t0);
1556 (square_2t0, x3, (#8.0));
1557 ((#4.0), x4, square_2t0);
1558 ((#4.0), x5, square_2t0);
1559 (square_2t0, x6, (#8.0))
1561 ( (dih_x x1 x2 x3 x4 x5 x6) <. (#1.381))`;;
1565 (* interval verification by Ferguson *)
1568 [((#4.0), x1, square_2t0);
1569 ((#4.0), x2, square_2t0);
1570 (square_2t0, x3, (#8.0));
1571 (square_2t0, x4, square_2t0);
1572 ((#4.0), x5, square_2t0);
1573 (square_2t0, x6, (#8.0))
1575 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#0.777))`;;
1580 (* interval verification by Ferguson *)
1583 [((#4.0), x1, square_2t0);
1584 ((#4.0), x2, square_2t0);
1585 (square_2t0, x3, (#8.0));
1586 (square_2t0, x4, square_2t0);
1587 ((#4.0), x5, square_2t0);
1588 (square_2t0, x6, (#8.0))
1590 ( (dih_x x1 x2 x3 x4 x5 x6) <. ( (#2.0) *. pi))`;;
1595 LOC: 2002 k.c page 45--46
1601 (* interval verification by Ferguson *)
1603 Uses monotonoicity in the x4 variable.
1607 [((#4.0), x1, square_2t0);
1608 ((#4.0), x2, square_2t0);
1609 ((#4.0), x3, square_2t0);
1610 ((#8.0), x4, (#8.0));
1611 ((#4.0), x5, square_2t0);
1612 ((#4.0), x6, square_2t0)
1615 ( (dih_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.372)) *. (sqrt x1)) +. ( (#0.465) *. (sqrt x2)) +. ( (#0.465) *. (sqrt x3)) +.
1616 ( (#0.465) *. (sqrt x5)) +. ( (#0.465) *. (sqrt x6))) >. (#4.885))`;;
1621 (* interval verification by Ferguson *)
1624 [((#4.0), x1, square_2t0);
1625 ((#4.0), x2, square_2t0);
1626 ((#4.0), x3, square_2t0);
1627 (square_2t0, x4, (#8.0));
1628 ((#4.0), x5, square_2t0);
1629 (square_2t0, x6, (#8.0))
1632 ( ( (#0.291) *. (sqrt x1)) +. ( (--. (#0.393)) *. (sqrt x2)) +. ( (--. (#0.586)) *. (sqrt x3)) +. ( (#0.79) *. (sqrt x4)) +.
1633 ( (--. (#0.321)) *. (sqrt x5)) +. ( (--. (#0.397)) *. (sqrt x6)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#2.47277)))`;;
1636 (* interval verification by Ferguson *)
1639 [((#4.0), x1, square_2t0);
1640 ((#4.0), x2, square_2t0);
1641 ((#4.0), x3, square_2t0);
1642 (square_2t0, x4, square_4t0);
1643 ((#4.0), x5, square_2t0);
1644 (square_2t0, x6, (#8.0))
1647 ( ( (#0.291) *. (sqrt x1)) +. ( (--. (#0.393)) *. (sqrt x2)) +. ( (--. (#0.586)) *. (sqrt x3)) +.
1648 ( (--. (#0.321)) *. (sqrt x5)) +. ( (--. (#0.397)) *. (sqrt x6)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#4.45567)))`;;
1652 (* interval verification by Ferguson *)
1655 [((#4.0), x1, square_2t0);
1656 ((#4.0), x2, square_2t0);
1657 ((#4.0), x3, square_2t0);
1658 ((#8.0), x4, square_4t0);
1659 ((#4.0), x5, square_2t0);
1660 (square_2t0, x6, (#8.0))
1663 ( ( (#0.291) *. (sqrt x1)) +. ( (--. (#0.393)) *. (sqrt x2)) +. ( (--. (#0.586)) *. (sqrt x3)) +.
1664 ( (--. (#0.321)) *. (sqrt x5)) +. ( (--. (#0.397)) *. (sqrt x6)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#4.71107)))`;;
1668 (* interval verification by Ferguson *)
1671 [((#4.0), x1, square_2t0);
1672 ((#4.0), x2, square_2t0);
1673 ((#4.0), x3, square_2t0);
1674 ((#8.0), x4, square_4t0);
1675 ((#4.0), x5, square_2t0);
1676 (square_2t0, x6, (#8.0))
1679 ( (--. (#0.214) *. (sqrt x1)) +. ( ( (#0.4)) *. (sqrt x2)) +. ( ( (#0.58)) *. (sqrt x3)) +.
1680 ( ( (#0.155)) *. (sqrt x5)) +. ( ( (#0.395)) *. (sqrt x6)) +. (dih_x x1 x2 x3 x4 x5 x6) ) >. (#4.52345))`;;
1684 (* interval verification in partK.cc *)
1687 [((#4.0), x1, square_2t0);
1688 ((#4.0), x2, square_2t0);
1689 ((#4.0), x3, square_2t0);
1690 ((#4.0), x4, square_2t0);
1691 (square_2t0, x5, (#8.0));
1692 (square_2t0, x6, (#8.0))
1694 ( (tauA_x x1 x2 x3 x4 x5 x6) >. D32)`;;
1697 (* interval verification in partK.cc *)
1700 [((#4.0), x1, square_2t0);
1701 ((#4.0), x2, square_2t0);
1702 ((#4.0), x3, square_2t0);
1703 ((#4.0), x4, square_2t0);
1704 (square_2t0, x5, (#8.0));
1705 (square_2t0, x6, (#8.0))
1707 ( (vorA_x x1 x2 x3 x4 x5 x6) <. Z32)`;;
1712 (* interval verification by Ferguson *)
1715 [((#4.0), x1, square_2t0);
1716 ((#4.0), x2, square_2t0);
1717 ((#4.0), x3, square_2t0);
1718 ((#4.0), x4, square_2t0);
1719 (square_2t0, x5, (#8.0));
1720 (square_2t0, x6, (#8.0))
1723 ( (( --. ) (sol_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.492)) *. (sqrt x1)) +. ( (--. (#0.492)) *. (sqrt x2)) +. ( (--. (#0.492)) *. (sqrt x3)) +.
1724 ( (#0.43) *. (sqrt x4)) +. ( (#0.038) *. (sqrt x5)) +. ( (#0.038) *. (sqrt x6)) ) <. (--. (#2.71884)))`;;
1728 (* interval verification in partK.cc *)
1731 [((#4.0), x1, square_2t0);
1732 ((#4.0), x2, square_2t0);
1733 ((#4.0), x3, square_2t0);
1734 ((#4.0), x4, square_2t0);
1735 (square_2t0, x5, (#8.0));
1736 (square_2t0, x6, (#8.0))
1739 ( (( --. ) (vorA_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.058)) *. (sqrt x1)) +. ( (--. (#0.105)) *. (sqrt x2)) +. ( (--. (#0.105)) *. (sqrt x3)) +.
1740 ( (--. (#0.115)) *. (sqrt x4)) +. ( (#0.062) *. (sqrt x5)) +. ( (--. (#0.062)) *. (sqrt x6)) ) >. (--. (#1.02014)))`;;
1744 (* interval verification in partK.cc *)
1747 [((#4.0), x1, square_2t0);
1748 ((#4.0), x2, square_2t0);
1749 ((#4.0), x3, square_2t0);
1750 ((#4.0), x4, square_2t0);
1751 (square_2t0, x5, (#8.0));
1752 (square_2t0, x6, (#8.0))
1755 ( (vor_0_x x1 x2 x3 x4 x5 x6) +. ( (#0.419351) *. (sol_x x1 x2 x3 x4 x5 x6)) ) <. (#0.3085))`;;
1759 (* interval verification by Ferguson *)
1762 [((#4.0), x1, square_2t0);
1763 ((#4.0), x2, square_2t0);
1764 ((#4.0), x3, square_2t0);
1765 ((#4.0), x4, square_2t0);
1766 (square_2t0, x5, (#8.0));
1767 (square_2t0, x6, (#8.0))
1770 ( ( (#0.115) *. (sqrt x1)) +. ( (--. (#0.452)) *. (sqrt x2)) +. ( (--. (#0.452)) *. (sqrt x3)) +.
1771 ( (#0.613) *. (sqrt x4)) +. ( (--. (#0.15)) *. (sqrt x5)) +. ( (--. (#0.15)) *. (sqrt x6)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#2.177)))`;;
1775 (* interval verification by Ferguson *)
1778 [((#4.0), x1, square_2t0);
1779 ((#4.0), x2, square_2t0);
1780 ((#4.0), x3, square_2t0);
1781 (square_2t0, x4, (#8.0));
1782 (square_2t0, x5, (#8.0));
1783 (square_2t0, x6, (#8.0))
1786 ( ( (#0.115) *. (sqrt x1)) +. ( (--. (#0.452)) *. (sqrt x2)) +. ( (--. (#0.452)) *. (sqrt x3)) +.
1787 ( (#0.618) *. (sqrt x4)) +. ( (--. (#0.15)) *. (sqrt x5)) +. ( (--. (#0.15)) *. (sqrt x6)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#2.17382)))`;;
1791 (* interval verification in partK.cc *)
1794 [((#4.0), x1, square_2t0);
1795 ((#4.0), x2, square_2t0);
1796 ((#4.0), x3, square_2t0);
1797 (square_2t0, x4, (#8.0));
1798 (square_2t0, x5, (#8.0));
1799 (square_2t0, x6, (#8.0))
1801 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.121)))`;;
1806 (* interval verification in partK.cc *)
1809 [((#4.0), x1, square_2t0);
1810 ((#4.0), x2, square_2t0);
1811 ((#4.0), x3, square_2t0);
1812 (square_2t0, x4, (#8.0));
1813 (square_2t0, x5, (#8.0));
1814 (square_2t0, x6, (#8.0))
1816 ( ((tau_0_x x1 x2 x3 x4 x5 x6)) >. (#0.21301))`;;
1819 (* interval verification by Ferguson *)
1822 [((#4.0), x1, square_2t0);
1823 ((#4.0), x2, square_2t0);
1824 ((#4.0), x3, square_2t0);
1825 (square_2t0, x4, square_4t0);
1826 (square_2t0, x5, (#8.0));
1827 (square_2t0, x6, (#8.0))
1830 ( ( (#0.115) *. (sqrt x1)) +. ( (--. (#0.452)) *. (sqrt x2)) +. ( (--. (#0.452)) *. (sqrt x3)) +.
1831 ( (--. (#0.15)) *. (sqrt x5)) +. ( (--. (#0.15)) *. (sqrt x6)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#3.725)))`;;
1836 (* interval verification by Ferguson *)
1839 [((#4.0), x1, square_2t0);
1840 ((#4.0), x2, square_2t0);
1841 ((#4.0), x3, square_2t0);
1842 ((#8.0), x4, square_4t0);
1843 (square_2t0, x5, (#8.0));
1844 (square_2t0, x6, (#8.0))
1847 ( ( (#0.115) *. (sqrt x1)) +. ( (--. (#0.452)) *. (sqrt x2)) +. ( (--. (#0.452)) *. (sqrt x3)) +.
1848 ( (--. (#0.15)) *. (sqrt x5)) +. ( (--. (#0.15)) *. (sqrt x6)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#3.927)))`;;
1854 LOC: 2002 k.c page 46
1861 [(square_2t0, x1, (#8.0));
1862 ((#4.0), x2, square_2t0);
1863 ((#4.0), x3, square_2t0);
1864 (square_2t0, x4, (#8.0));
1865 ((#4.0), x5, square_2t0);
1866 ((#4.0), x6, square_2t0)
1868 ( (vorC_x x1 x2 x3 x4 x5 x6) <. (#0.0))`;;
1875 [(square_2t0, x1, (#8.0));
1876 ((#4.0), x2, square_2t0);
1877 ((#4.0), x3, square_2t0);
1878 (square_2t0, x4, (#8.0));
1879 ((#4.0), x5, square_2t0);
1880 ((#4.0), x6, square_2t0)
1883 ( ( (#0.47) *. (sqrt x1)) +. ( (--. (#0.522)) *. (sqrt x2)) +. ( (--. (#0.522)) *. (sqrt x3)) +. ( (#0.812) *. (sqrt x4)) +.
1884 ( (--. (#0.522)) *. (sqrt x5)) +. ( (--. (#0.522)) *. (sqrt x6)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#2.82988)))`;;
1888 (* Uses monotonicity in the x4 variable *)
1892 [(square_2t0, x1, (#8.0));
1893 ((#4.0), x2, square_2t0);
1894 ((#4.0), x3, square_2t0);
1895 (square_2t0, x4, square_2t0);
1896 ((#4.0), x5, square_2t0);
1897 ((#4.0), x6, square_2t0)
1900 ( ( (#0.47) *. (sqrt x1)) +. ( (--. (#0.522)) *. (sqrt x2)) +. ( (--. (#0.522)) *. (sqrt x3)) +.
1901 ( (--. (#0.522)) *. (sqrt x5)) +. ( (--. (#0.522)) *. (sqrt x6)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#4.8681)))`;;
1905 (* Uses monotonicity in x4 *)
1909 [(square_2t0, x1, (#8.0));
1910 ((#4.0), x2, square_2t0);
1911 ((#4.0), x3, square_2t0);
1912 ((#8.0), x4, (#8.0));
1913 ((#4.0), x5, square_2t0);
1914 ((#4.0), x6, square_2t0)
1917 ( ( (#0.47) *. (sqrt x1)) +. ( (--. (#0.522)) *. (sqrt x2)) +. ( (--. (#0.522)) *. (sqrt x3)) +.
1918 ( (--. (#0.522)) *. (sqrt x5)) +. ( (--. (#0.522)) *. (sqrt x6)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#5.1623)))`;;
1925 LOC: 2002 k.c page 47
1933 [((#4.0), x1, square_2t0);
1934 ((#4.0), x2, square_2t0);
1935 (square_2t0, x3, (#8.0));
1936 (square_2t0, x4, square_4t0);
1937 ((#4.0), x5, square_2t0);
1938 ((#4.0), x6, square_2t0)
1941 ( ( (--. (#0.4)) *. (sqrt x3)) +. ( (#0.15) *. (sqrt x1)) +. ( (--. (#0.09)) *. (sqrt x2)) +.
1942 ( (--. (#0.631)) *. (sqrt x6)) +. ( (--. (#0.23)) *. (sqrt x5)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#3.9788)))`;;
1948 [((#4.0), x1, square_2t0);
1949 ((#4.0), x2, square_2t0);
1950 (square_2t0, x3, (#8.0));
1951 ((#4.0), x4, square_2t0);
1952 ((#4.0), x5, square_2t0);
1953 (square_2t0, x6, (#8.0))
1956 ( ( (#0.289) *. (sqrt x1)) +. ( (--. (#0.148)) *. (sqrt x2)) +. ( (--. (#1.36)) *. (sqrt x3)) +.
1957 ( (#0.688) *. (sqrt x4)) +. ( (--. (#0.148)) *. (sqrt x5)) +. ( (--. (#1.36)) *. (sqrt x6)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#6.3282)))`;;
1964 [((#4.0), x1, square_2t0);
1965 ((#4.0), x2, square_2t0);
1966 (square_2t0, x3, (#8.0));
1967 (square_2t0, x4, (square (( +. ) (#2.51) (sqrt (#8.0)))));
1968 ((#4.0), x5, square_2t0);
1969 (square_2t0, x6, (#8.0))
1972 ( ( (#0.289) *. (sqrt x1)) +. ( (--. (#0.148)) *. (sqrt x2)) +. ( (--. (#0.723)) *. (sqrt x3)) +.
1973 ( (--. (#0.148)) *. (sqrt x5)) +. ( (--. (#0.723)) *. (sqrt x6)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) ) <. (--. (#4.85746)))`;;
1980 LOC: 2002 k.c page 47
1985 (* interval verification in partK.cc *)
1988 [((square (#2.696)), x1, (#8.0));
1989 ((square (#2.45)), x2, square_2t0);
1990 ((#4.0), x3, square_2t0);
1991 ((#4.0), x4, square_2t0);
1992 ((#4.0), x5, square_2t0);
1993 ((square (#2.45)), x6, square_2t0)
1995 ( (nu_x x1 x2 x3 x4 x5 x6) <. (--. (#0.055)))`;;
2001 [((square (#2.696)), x1, (#8.0));
2002 ((square (#2.45)), x2, square_2t0);
2003 ((#4.0), x3, square_2t0);
2004 ((#4.0), x4, square_2t0);
2005 ((#4.0), x5, square_2t0);
2006 ((square (#2.45)), x6, square_2t0)
2008 ( (taunu_x x1 x2 x3 x4 x5 x6) >. (#0.092))`;;
2014 (* interval verification in partK.cc *)
2017 [((#4.0), x1, square_2t0);
2018 ((square (#2.45)), x2, square_2t0);
2019 ((#4.0), x3, square_2t0);
2020 (square_2t0, x4, (#8.0));
2021 ((#4.0), x5, square_2t0);
2022 ((#4.0), x6, square_2t0)
2024 ( (sigmahat_x x1 x2 x3 x4 x5 x6) <. (--. (#0.039)))`;;
2028 [((#4.0), x1, square_2t0);
2029 ((square (#2.45)), x2, square_2t0);
2030 ((#4.0), x3, square_2t0);
2031 (square_2t0, x4, (#8.0));
2032 ((#4.0), x5, square_2t0);
2033 ((#4.0), x6, square_2t0)
2035 ( (tauhat_x x1 x2 x3 x4 x5 x6) >. (#0.094))`;;
2037 (* interval verification in partK.cc *)
2040 [((square (#2.696)), x1, (#8.0));
2041 ((square (#2.45)), x2, square_2t0);
2042 ((#4.0), x3, square_2t0);
2043 (square_2t0, x4, (#8.0));
2044 ((#4.0), x5, square_2t0);
2045 ((square (#2.45)), x6, square_2t0)
2047 ( (vor_analytic_x x1 x2 x3 x4 x5 x6) <. (--. (#0.197)))`;;
2053 [((square (#2.696)), x1, (#8.0));
2054 ((square (#2.45)), x2, square_2t0);
2055 ((#4.0), x3, square_2t0);
2056 (square_2t0, x4, (#8.0));
2057 ((#4.0), x5, square_2t0);
2058 ((square (#2.45)), x6, square_2t0)
2060 ( (taunu_x x1 x2 x3 x4 x5 x6) >. (#0.239))`;;
2066 (* interval verification by Ferguson *)
2067 (* interval verification by Ferguson *)
2070 [((square (#2.45)), x1, square_2t0);
2071 ((#4.0), x2, square_2t0);
2072 ((#4.0), x3, square_2t0);
2073 ((#4.0), x4, square_2t0);
2075 (square_2t0, x5, (#8.0));
2076 (square_2t0, x6, (#8.0))
2078 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.089)))`;;
2084 [((square (#2.45)), x1, square_2t0);
2085 ((#4.0), x2, square_2t0);
2086 ((#4.0), x3, square_2t0);
2087 ((#4.0), x4, square_2t0);
2089 (square_2t0, x5, (#8.0));
2090 (square_2t0, x6, (#8.0))
2092 ( (tau_0_x x1 x2 x3 x4 x5 x6) >. (#0.154))`;;
2096 (* interval verification by Ferguson *)
2099 [((square (#2.45)), x1, square_2t0);
2100 ((#4.0), x2, square_2t0);
2101 ((#4.0), x3, square_2t0);
2102 (square_2t0, x4, (#8.0));
2104 (square_2t0, x5, (#8.0));
2105 ((#4.0), x6, square_2t0)
2107 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.089)))`;;
2113 [((square (#2.45)), x1, square_2t0);
2114 ((#4.0), x2, square_2t0);
2115 ((#4.0), x3, square_2t0);
2116 (square_2t0, x4, (#8.0));
2118 (square_2t0, x5, (#8.0));
2119 ((#4.0), x6, square_2t0)
2121 ( (tau_0_x x1 x2 x3 x4 x5 x6) >. (#0.154))`;;
2126 LOC: 2002 k.c page 47
2131 (* interval verification in partK.cc *)
2134 [(square_2t0, x1, (square (#2.696)));
2135 ((#4.0), x2, square_2t0);
2136 ((#4.0), x3, square_2t0);
2137 ((#4.0), x4, square_2t0);
2139 ((#4.0), x5, square_2t0);
2140 ((#4.0), x6, square_2t0)
2143 ( (octavor_analytic_x x1 x2 x3 x4 x5 x6) <. ( (octavor0_x x1 x2 x3 x4 x5 x6) +. (--. (#0.017)))) \/
2144 ( (eta_x x1 x2 x6) <. (sqrt (#2.0))))`;;
2149 (* interval verification in partK.cc *)
2152 [(square_2t0, x1, (square (#2.696)));
2153 ((#4.0), x2, square_2t0);
2154 ((#4.0), x3, square_2t0);
2155 ((#9.0), x4, (#9.0));
2157 ((#4.0), x5, square_2t0);
2158 ((#4.0), x6, square_2t0)
2161 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.678)) \/
2162 ( ( (sqrt x2) +. (sqrt x3) +. (sqrt x5) +. (sqrt x6)) >. (#8.77)))`;;
2167 LOC: 2002 k.c page 47
2173 (* interval verification in partK.cc *)
2176 [((#4.0), x1, square_2t0);
2177 ((#4.0), x2, square_2t0);
2178 ((#4.0), x3, square_2t0);
2179 (square_2t0, x4, (square (#2.6)));
2181 ((#4.0), x5, (square (#2.138)));
2182 ((#4.0), x6, square_2t0)
2184 ( (gamma_x x1 x2 x3 x4 x5 x6) <. ( (#0.3138) +. ( (--. (#0.157)) *. (sqrt x5))))`;;
2189 (* interval verification in partK.cc *)
2192 [((#4.0), x1, square_2t0);
2193 ((square (#2.121)), x2, (square (#2.145)));
2194 ((#4.0), x3, square_2t0);
2195 (square_2t0, x4, (#8.0));
2197 ((square (#2.22)), x5, (square (#2.238)));
2198 ((#4.0), x6, square_2t0)
2200 ( (gamma_x x1 x2 x3 x4 x5 x6) <. (--. (#0.06)))`;;
2206 Earlier version was false at (4.0,4.0,4.0,4.0,5.5225,5.5225).
2207 Bug fixed 1/19/2008 : lower bound on x4 was a typo. It should be square_2t0.
2209 (* interval verification in partK.cc *)
2212 [((#4.0), x1, (square (#2.2)));
2213 ((#4.0), x2, (square (#2.2)));
2214 ((#4.0), x3, (square (#2.2)));
2215 (square_2t0, x4, (#8.0));
2216 ((#4.0), x5, (square (#2.35)));
2217 ((#4.0), x6, (square (#2.35)))
2219 ( (gamma_x x1 x2 x3 x4 x5 x6) <.
2220 ( (#0.000001) +. (#1.4) +. ( (--. (#0.1)) *. (sqrt x1))
2221 +. ( (--. (#0.15)) *. ( (sqrt x2) +. (sqrt x3) +.
2222 (sqrt x5) +. (sqrt x6)))))`;;
2229 LOC: 2002 k.c page 48
2233 (* interval verification in partK.cc *)
2236 [((#4.0), x1, (square (#2.14)));
2237 ((#4.0), x2, (square (#2.14)));
2238 ((#4.0), x3, (square (#2.14)));
2239 ((square (#2.7)), x4, (#8.0));
2241 ((#4.0), x5, square_2t0);
2242 ((#4.0), x6, square_2t0)
2244 ( (vor_analytic_x x1 x2 x3 x4 x5 x6) <. (vor_0_x x1 x2 x3 x4 x5 x6))`;;
2249 (* interval verification in partK.cc *)
2252 [((#4.0), x1, square_2t0);
2253 ((#4.0), x2, square_2t0);
2254 ((#4.0), x3, square_2t0);
2255 (square_2t0, x4, (square (#2.72)));
2257 ((#4.0), x5, square_2t0);
2258 ((#4.0), x6, square_2t0)
2261 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.064))) \/
2262 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
2266 (* interval verification in partK.cc *)
2269 [((#4.0), x1, square_2t0);
2270 ((#4.0), x2, square_2t0);
2271 ((#4.0), x3, square_2t0);
2272 ((square (#2.7)), x4, (#8.0));
2274 ((#4.0), x5, square_2t0);
2275 ((#4.0), x6, square_2t0)
2277 ( (vor_0_x x1 x2 x3 x4 x5 x6) <.
2278 ( (#1.0612) +. ( (--. (#0.08)) *. ( (sqrt x1) +. (sqrt x2) +. (sqrt x3))) +. ( (--. (#0.142)) *. ( (sqrt x5) +. (sqrt x6)))))`;;
2284 SphereIn[5]:= VorVc @@ Sqrt [{4,4,4,6.7081,6.1009,4.41}]
2285 SphereOut[5]= -0.0625133.
2286 1/19/2008. Added the missing eta456 constraint to eliminate counterexample.
2288 (* interval verification in partK.cc *)
2291 [((#4.0), x1, square_2t0);
2292 ((#4.0), x2, square_2t0);
2293 ((#4.0), x3, square_2t0);
2294 ((square (#2.59)), x4, (square (#2.64)));
2295 ((square (#2.47)), x5, square_2t0);
2296 ((square (#2.1)), x6, (square (#3.51)))
2298 (( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.0713))) \/
2299 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
2305 [((#4.0), x1, (square (#2.13)));
2306 ((#4.0), x2, (square (#2.13)));
2307 ((#4.0), x3, (square (#2.13)));
2308 ((square (#2.7)), x4, (square (#2.74)));
2310 ((#4.0), x5, square_2t0);
2311 ((#4.0), x6, square_2t0)
2314 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.06))) \/
2315 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
2320 (* interval verification in partK.cc *)
2323 [((#4.0), x1, square_2t0);
2324 ((#4.0), x2, square_2t0);
2325 ((#4.0), x3, square_2t0);
2326 (square_2t0, x4, (square (#2.747)));
2328 ((#4.0), x5, square_2t0);
2329 ((#4.0), x6, square_2t0)
2332 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.058))) \/
2333 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
2338 (* interval verification in partK.cc *)
2341 [((#4.0), x1, square_2t0);
2342 ((#4.0), x2, square_2t0);
2343 ((#4.0), x3, square_2t0);
2344 (square_2t0, x4, (square (#2.77)));
2346 ((#4.0), x5, square_2t0);
2347 ((#4.0), x6, square_2t0)
2350 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.0498))) \/
2351 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
2357 LOC: 2002 k.c page 48
2364 Was false at (4,4,4,8,6.3001,6.3001)
2365 Fixed by inserting the missing circumradius condition on 1/19/2008.
2366 Also, the lower bound on x4 was changed to 7.29 from square_2t0
2367 to bring it into agreement with the interval calculation in partK.cc
2369 (* interval verification in partK.cc *)
2371 (* changed (square_2t0, x4, (#8.0)); *)
2375 [((#4.0), x1, (square (#2.14)));
2376 ((#4.0), x2, (square (#2.14)));
2377 ((#4.0), x3, (square (#2.14)));
2378 ((square (#7.29), x4, (#8.0)));
2379 ((#4.0), x5, square_2t0);
2380 ((#4.0), x6, square_2t0)
2382 (( ( (( --. ) (gamma_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.145)) *. (sqrt x1)) +. ( (--. (#0.08)) *. ( (sqrt x2) +. (sqrt x3))) +.
2383 ( (--. (#0.133)) *. ( (sqrt x5) +. (sqrt x6)))) >. (--. (#1.146))) \/ ( (eta_x x4 x5 x6) >. (sqrt (#2.0))))`;;
2387 (* interval verification in partK.cc *)
2390 point: [4, 4, 4, 6.3001, 5.29, 5.29]
2391 value: about 0.0001.
2393 The interval arithmetic code for 381970727 in partK.c has a lower
2394 bound on x4 of 7.29. This seems to be a bug in the 1998 interval arithmetic
2395 code. A note has been added to the dcg_errata.
2396 This affects the 1998 linear programs.
2398 I am changing the lower bound on x4 to 7.29, even though we would like
2399 it to be at its original 6.3001. TCH 1/29/2008.
2402 (* Please don't put comments inside HOL terms. They don't compile. Oh no! *)
2403 (* let I_381970727= *)
2404 (* all_forall `ineq *)
2405 (* [((#4.0), x1, (square (#2.14))); *)
2406 (* ((#4.0), x2, (square (#2.14))); *)
2407 (* ((#4.0), x3, (square (#2.14))); *)
2408 (* (\* (square_2t0, x4, (#8.0)); *\) *)
2409 (* ((#7.29), x4, (#8.0)); *)
2411 (* ((#4.0), x5, (square (#2.3))); *)
2412 (* ((#4.0), x6, (square (#2.3))) *)
2414 (* ( ( (( --. ) (gamma_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.145)) *. (sqrt x1)) +. ( (--. (#0.081)) *. ( (sqrt x2) +. (sqrt x3))) +. *)
2415 (* ( (--. (#0.16)) *. ( (sqrt x5) +. (sqrt x6)))) >. (--. (#1.255)))`;; *)
2419 [((#4.0), x1, (square (#2.14)));
2420 ((#4.0), x2, (square (#2.14)));
2421 ((#4.0), x3, (square (#2.14)));
2422 ((#7.29), x4, (#8.0));
2423 ((#4.0), x5, (square (#2.3)));
2424 ((#4.0), x6, (square (#2.3)))
2426 ( ( (( --. ) (gamma_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.145)) *. (sqrt x1)) +. ( (--. (#0.081)) *. ( (sqrt x2) +. (sqrt x3))) +.
2427 ( (--. (#0.16)) *. ( (sqrt x5) +. (sqrt x6)))) >. (--. (#1.255)))`;;
2429 (* interval verification in partK.cc *)
2431 (* This was false at
2432 point: [4, 4, 4, 8, 4, 4]
2434 However, this doesn't satisfy the second constraint:
2435 In the paper and in partK.cc, there is a constraint that y5+y6 >= 4.3.
2436 This was overlooked when this inequality was originally typed.
2437 This fixes the problem.
2441 [((#4.0), x1, (square (#2.14)));
2442 ((#4.0), x2, (square (#2.14)));
2443 ((#4.0), x3, (square (#2.14)));
2444 (square_2t0, x4, (#8.0));
2445 ((#4.0), x5, (square (#2.3)));
2446 ((#4.0), x6, (square (#2.3)))
2448 ((((( --. ) (gamma_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.03)) *. (sqrt x1)) +. ( (--. (#0.03)) *. ( (sqrt x2) +. (sqrt x3))) +.
2449 ( (--. (#0.094)) *. ( (sqrt x5) +. (sqrt x6)))) >. (--. (#0.5361)))
2450 \/ ((sqrt x5) +. (sqrt x6) <. #4.3))`;;
2452 (* interval verification in partK.cc *)
2455 [((#4.0), x1, (square (#2.14)));
2456 ((#4.0), x2, (square (#2.14)));
2457 ((#4.0), x3, (square (#2.14)));
2458 (square_2t0, x4, (#8.0));
2460 ((#4.0), x5, square_2t0);
2461 ((#4.0), x6, square_2t0)
2464 ( ( (( --. ) (gamma_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.03)) *. (sqrt x1)) +. ( (--. (#0.03)) *. ( (sqrt x2) +. (sqrt x3))) +.
2465 ( (--. (#0.16)) *. ( (sqrt x5) +. (sqrt x6)))) >. ( (--. (#0.82)) +. (--. (#0.000001)))) \/
2466 ( ( (sqrt x5) +. (sqrt x6)) >. (#4.3)))`;;
2471 (* interval verification in partK.cc *)
2473 This was false. Gamma seems unstable as
2477 point: {4, 4, 4, 7.99999999999, 15.3886219273, 6.30009999999}]
2479 Typo fixed on the upper bound of x5.
2480 The correct upper bound square_2t0.
2485 [((#4.0), x1, (square (#2.14)));
2486 ((#4.0), x2, (square (#2.14)));
2487 ((#4.0), x3, (square (#2.14)));
2488 (square_2t0, x4, (#8.0));
2489 ((square (#2.35)), x5, square_2t0);
2490 ((#4.0), x6, square_2t0)
2492 ( (gamma_x x1 x2 x3 x4 x5 x6) <. (--. (#0.053)))`;;
2496 (* interval verification in partK.cc *)
2499 [((#4.0), x1, (square (#2.14)));
2500 ((#4.0), x2, (square (#2.14)));
2501 ((#4.0), x3, (square (#2.14)));
2502 (square_2t0, x4, (#8.0));
2504 ((square (#2.25)), x5, square_2t0);
2505 ((#4.0), x6, square_2t0)
2507 ( (gamma_x x1 x2 x3 x4 x5 x6) <. (--. (#0.041)))`;;
2512 (* interval verification in partK.cc *)
2515 [((#4.0), x1, (square (#2.14)));
2516 ((#4.0), x2, (square (#2.14)));
2517 ((#4.0), x3, (square (#2.14)));
2518 (square_2t0, x4, (#8.0));
2520 ((#4.0), x5, square_2t0);
2521 ((#4.0), x6, square_2t0)
2524 ( ( (gamma_x x1 x2 x3 x4 x5 x6) +. ( (#0.419351) *. (sol_x x1 x2 x3 x4 x5 x6))) <.
2525 ( ( (#0.079431) *. (dih_x x1 x2 x3 x4 x5 x6)) +. ( (#0.0436) *. ( (sqrt x5) +. (sqrt x6))) +. (--. (#0.0294)))) \/
2526 ( (eta_x x4 x5 x6) >. (sqrt (#2.0))))`;;
2530 (* interval verification in partK.cc *)
2533 [((#4.0), x1, (square (#2.13)));
2534 ((#4.0), x2, (square (#2.13)));
2535 ((#4.0), x3, (square (#2.13)));
2536 (square_2t0, x4, (square (#2.67)));
2538 ((#4.0), x5, (square (#2.1)));
2539 ((square (#2.27)), x6, (square (#2.43)))
2541 ( (gamma_x x1 x2 x3 x4 x5 x6) <. ( (#1.1457) +. ( (--. (#0.1)) *. ( (sqrt x1) +. (sqrt x2) +. (sqrt x3))) +.
2542 ( (--. (#0.17)) *. (sqrt x5)) +. ( (--. (#0.11)) *. (sqrt x6))))`;;
2547 (* interval verification in partK.cc *)
2550 [((#4.0), x1, (square (#2.14)));
2551 ((#4.0), x2, (square (#2.14)));
2552 ((#4.0), x3, (square (#2.14)));
2553 (square_2t0, x4, (square (#2.7)));
2555 ((#4.0), x5, square_2t0);
2556 ((#4.0), x6, square_2t0)
2559 ( ( ( (#1.69) *. (sqrt x4)) +. (sqrt x5) +. (sqrt x6)) >. (#9.0659)) \/
2560 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
2564 (* interval verification in partK.cc *)
2567 [((#4.0), x1, (square (#2.14)));
2568 ((#4.0), x2, (square (#2.14)));
2569 ((#4.0), x3, (square (#2.14)));
2570 (square_2t0, x4, (square (#2.77)));
2572 ((#4.0), x5, square_2t0);
2573 ((#4.0), x6, square_2t0)
2576 ( ( ( (#1.69) *. (sqrt x4)) +. (sqrt x5) +. (sqrt x6)) >. (#9.044)) \/
2577 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
2582 (* interval verification in partK.cc *)
2585 [((#4.0), x1, (square (#2.14)));
2586 ((#4.0), x2, (square (#2.14)));
2587 ((#4.0), x3, (square (#2.14)));
2588 (square_2t0, x4, (square (#2.72)));
2590 ((#4.0), x5, square_2t0);
2591 ((#4.0), x6, square_2t0)
2594 ( ( (sqrt x5) +. (sqrt x6)) >. (#4.4)) \/
2595 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
2601 LOC: 2002 k.c page 48
2607 (* interval verification in partK.cc *)
2610 [((#4.0), x1, (square (#2.14)));
2611 ((#4.0), x2, (square (#2.14)));
2612 ((#4.0), x3, (square (#2.14)));
2613 ((#4.0), x4, square_2t0);
2614 (square_2t0, x5, (square (#2.77)));
2615 (square_2t0, x6, (square (#2.77)))
2617 ( ( (( --. ) (vor_analytic_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.058)) *. (sqrt x1)) +. ( (--. (#0.08)) *. (sqrt x2)) +. ( (--. (#0.08)) *. (sqrt x3)) +.
2618 ( (--. (#0.16)) *. (sqrt x4)) +. ( (--. (#0.21)) *. ( (sqrt x5) +. (sqrt x6))) ) >. (--. (#1.7531)))`;;
2623 Changed x5 from 4..(2t)^2
2625 (* interval verification in partK.cc *)
2628 [((#4.0), x1, (square (#2.14)));
2629 ((#4.0), x2, (square (#2.14)));
2630 ((#4.0), x3, (square (#2.14)));
2631 ((#4.0), x4, square_2t0);
2632 (square_2t0, x5, #8.0);
2633 ((square (#2.77)), x6, (#8.0))
2636 ( ( (( --. ) (vor_0_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.058)) *. (sqrt x1)) +. ( (--. (#0.1)) *. (sqrt x2)) +. ( (--. (#0.1)) *. (sqrt x3)) +.
2637 ( (--. (#0.165)) *. (sqrt x4)) +. ( (--. (#0.115)) *. (sqrt x6)) +. ( (--. (#0.12)) *. (sqrt x5)) ) >. (--. (#1.38875))) \/
2638 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
2643 (* interval verification in partK.cc *)
2646 [((#4.0), x1, square_2t0);
2647 ((#4.0), x2, square_2t0);
2648 ((#4.0), x3, square_2t0);
2649 ((#4.0), x4, square_2t0);
2651 (square_2t0, x5, (square (#2.77)));
2652 (square_2t0, x6, (square (#2.77)))
2655 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) >. (#7.206)) \/
2656 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
2660 (* interval verification in partK.cc *)
2663 [((#4.0), x1, (square (#2.14)));
2664 ((#4.0), x2, (square (#2.14)));
2665 ((#4.0), x3, (square (#2.14)));
2666 ((#4.0), x4, square_2t0);
2668 (square_2t0, x5, (square (#2.77)));
2669 (square_2t0, x6, (square (#2.77)))
2672 ( ( (( --. ) (vor_0_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.058)) *. (sqrt x1)) +. ( (--. (#0.05)) *. (sqrt x2)) +. ( (--. (#0.05)) *. (sqrt x3)) +.
2673 ( (--. (#0.16)) *. (sqrt x4)) +. ( (--. (#0.13)) *. (sqrt x6)) +. ( (--. (#0.13)) *. (sqrt x5)) ) >. (--. (#1.24547))) \/
2674 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
2680 (* interval verification in partK.cc *)
2683 [((#4.0), x1, square_2t0);
2684 ((#4.0), x2, square_2t0);
2685 ((#4.0), x3, square_2t0);
2686 (square_2t0, x4, (#8.0));
2688 ((square (#2.77)), x5, (#8.0));
2689 ((#4.0), x6, square_2t0)
2691 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.077)))`;;
2695 (* interval verification in partK.cc *)
2698 [((#4.0), x1, square_2t0);
2699 ((#4.0), x2, square_2t0);
2700 ((#4.0), x3, square_2t0);
2701 (square_2t0, x4, (square (#2.77)));
2702 (square_2t0, x5, (square (#2.77)));
2703 ((#4.0), x6, square_2t0)
2705 ( ( (vor_analytic_x x1 x2 x3 x4 x5 x6) +. ( (#0.419351) *. (sol_x x1 x2 x3 x4 x5 x6))) <. (#0.289))`;;
2708 (* I_900212351 has been deprecated. *)
2712 LOC: 2002 k.c page 49
2719 (* interval verification in partK.cc *)
2722 [(square_2t0, x1, (#8.0));
2723 ((#4.0), x2, square_2t0);
2724 ((#4.0), x3, square_2t0);
2725 (square_2t0, x4, (square (#2.6961)));
2727 ((#4.0), x5, square_2t0);
2728 ((#4.0), x6, square_2t0)
2730 ( (vor_analytic_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.078)) / (#2.0)))`;;
2735 (* interval verification in partK.cc *)
2738 [((#4.0), x1, square_2t0);
2739 ((#4.0), x2, square_2t0);
2740 ((#4.0), x3, square_2t0);
2741 ((#4.0), x4, square_2t0);
2743 (square_2t0, x5, (square (#2.6961)));
2744 (square_2t0, x6, (square (#2.6961)))
2747 ( (vort_x x1 x2 x3 x4 x5 x6 (sqrt (#2.0))) <. (--. (#0.078))) \/
2748 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
2755 LOC: 2002 k.c page 49
2764 The interval calculations here show that the set of separated
2765 vertices (\ref{definition:admissible:excess}) can be generalized
2766 to include opposite vertices of a quadrilateral unless the edge
2767 between those vertices forms a flat quarter. Consider a vertex
2768 of type $(3,1,1)$ with $a(3)=1.4\,\pt$. By the arguments in the
2769 text, we may assume that the dihedral angles of the exceptional
2770 regions at those vertices are at least $1.32$ (see
2771 \cite[3.11.4]{part4}). Also, the three quasi-regular tetrahedra at
2772 the vertex squander at least $1.5\,\pt$ by a linear programming
2773 bound, if the angle of the quad cluster is at least $1.55$. Thus,
2774 we assume that the dihedral angles at opposite vertices of the
2775 quad cluster are at most $1.55$. A linear program also gives
2776 $\tau+0.316\dih>0.3864$ for a quasi-regular tetrahedron.
2778 If we give bounds of the form
2779 $\tau_x +0.316\dih> b$, for the part of the quad cluster around a
2780 vertex, where $\tau_x$ is the appropriate squander function, then
2782 $$\sum\tau_x > -0.316(2\pi-1.32) + b + 3 (0.3864)$$
2783 for a lower bound on what is squandered. If the two opposite
2784 vertices give at least $2(1.4)\,\pt + 0.1317$, then the inclusion
2785 of two opposite vertices in the separated set of vertices is
2786 justified. (Recall that $t_4=0.1317$.) The following
2787 inequalities give the desired result.
2793 (* interval verification in partK.cc *)
2796 [((#4.0), x1, square_2t0);
2797 ((#4.0), x2, square_2t0);
2798 ((#4.0), x3, square_2t0);
2799 (square_2t0, x4, (#8.0));
2801 ((#4.0), x5, square_2t0);
2802 ((#4.0), x6, square_2t0)
2805 ( ( (taumu_flat_x x1 x2 x3 x4 x5 x6) +. ( (#0.316) *. (dih_x x1 x2 x3 x4 x5 x6))) >. (#0.5765)) \/
2806 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.55)))`;;
2810 (* interval verification in partK.cc *)
2812 CCC fails: added delta > 0
2816 [((#4.0), x1, square_2t0);
2817 ((#4.0), x2, square_2t0);
2818 ((#4.0), x3, square_2t0);
2819 ((#8.0), x4, square_4t0);
2820 ((#4.0), x5, square_2t0);
2821 ((#4.0), x6, square_2t0)
2824 ( ( (tau_0_x x1 x2 x3 x4 x5 x6) +. ( (#0.316) *. (dih_x x1 x2 x3 x4 x5 x6))) >. (#0.5765)) \/
2825 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.55)) \/
2826 ((delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
2831 (* interval verification in partK.cc *)
2834 [(square_2t0, x1, (#8.0));
2835 ((#4.0), x2, square_2t0);
2836 ((#4.0), x3, square_2t0);
2837 ((#4.0), x4, square_2t0);
2839 ((#4.0), x5, square_2t0);
2840 ((#4.0), x6, square_2t0)
2842 ( ( (taunu_x x1 x2 x3 x4 x5 x6) +. ( (#0.316) *. (dih2_x x1 x2 x3 x4 x5 x6))) >. (#0.2778))`;;
2845 (* Note I moved the huge non-interval-arithmetic inequalitites
2846 to kep_inequalities2.ml *)
2850 LOC: 2002 k.c page 50
2855 (* interval verification in partK.cc *)
2858 [(square_2t0, x1, (#8.0));
2859 ((#4.0), x2, square_2t0);
2860 ((#4.0), x3, square_2t0);
2861 (square_2t0, x4, (#8.0));
2863 ((#4.0), x5, square_2t0);
2864 ((#4.0), x6, square_2t0)
2866 ( (vorC_x x1 x2 x3 x4 x5 x6) <. (#0.0))`;;
2873 [(square_2t0, x1, (square (#2.696)));
2874 ((#4.0), x2, square_2t0);
2875 ((#4.0), x3, square_2t0);
2876 (square_2t0, x4, (#8.0));
2878 ((#4.0), x5, square_2t0);
2879 ((#4.0), x6, square_2t0)
2881 ( (vorC_x x1 x2 x3 x4 x5 x6) <. (--. (#0.05)))`;;
2888 [(square_2t0, x1, (square (#2.696)));
2889 ((#4.0), x2, square_2t0);
2890 ((#4.0), x3, square_2t0);
2891 (square_2t0, x4, (#8.0));
2893 ((#4.0), x5, square_2t0);
2894 ((#4.0), x6, square_2t0)
2897 ( (vorC_x x1 x2 x3 x4 x5 x6) <. (--. (#0.119))) \/
2898 ( (eta_x x1 x2 x6) <. (sqrt (#2.0))))`;;
2903 LOC: 2002 k.c page 50-51
2914 0,& y_1\in[2t_0,2\sqrt2],\\
2915 -0.043, & y_1\in[2t_0,2.696],\\
2920 for quad regions $R$ constructed from an anchored
2921 simplex $S$ and adjacent special simplex $S'$. Assume that
2922 $y_4(S)=y_4(S')\in[2\sqrt2,3.2]$, and that the other edges have
2923 lengths in $[2,2t_0]$. The bound $0$ is found in \cite[Lemma
2924 3.13]{formulation}. The bound $-0.043$ is obtained from
2925 deformations, reducing the inequality to the following interval
2928 (* interval verification by Ferguson *)
2929 \refno{368244553\dag}
2931 (* interval verification by Ferguson *)
2932 \refno{820900672\dag}
2934 (* interval verification by Ferguson *)
2935 \refno{961078136\dag}
2938 Under certain conditions, Inequality \ref {eqn:group24} can be
2941 (The last of these was verified by S. Ferguson.) \refno{424186517}
2943 These combine to give
2945 \vor_0(S)+\vor_0(S') < \begin{cases} -0.091,&\text{ or }\\
2949 for the combination of special simplex and anchored simplex under
2950 the stated conditions.
2957 [(square_2t0, x1, (square (#2.696)));
2958 ((#4.0), x2, square_2t0);
2959 ((#4.0), x3, square_2t0);
2960 ((#4.0), x4, square_2t0);
2962 ((#4.0), x5, square_2t0);
2963 (square_2t0, x6, square_2t0)
2965 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.043)) / (#2.0)))`;;
2969 [(square_2t0, x1, (square (#2.696)));
2970 ((#4.0), x2, square_2t0);
2971 ((#4.0), x3, square_2t0);
2972 ((#8.0), x4, (square (#3.2)));
2974 ((#4.0), x5, square_2t0);
2975 ((#4.0), x6, square_2t0)
2978 ( ( (vor_0_x x1 x2 x3 x4 x5 x6) +. (vor_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))) <. (--. (#0.043))) \/
2979 ( (cross_diag_x x1 x2 x3 x4 x5 x6 (#4.0) (#4.0) (#4.0)) <. two_t0))`;;
2983 [(square_2t0, x1, (square (#2.696)));
2984 ((#4.0), x2, square_2t0);
2985 ((#4.0), x3, square_2t0);
2986 ((#8.0), x4, (square (#3.2)));
2988 ((#4.0), x5, square_2t0);
2989 ((#4.0), x6, square_2t0)
2992 ( ( (vor_0_x x1 x2 x3 x4 x5 x6) +. (vor_0_x square_2t0 x2 x3 x4 (#4.0) (#4.0))) <. (--. (#0.043))) \/
2993 ( (cross_diag_x x1 x2 x3 x4 x5 x6 square_2t0 (#4.0) (#4.0)) <. two_t0))`;;
2995 (* Fixed bad bounds on first variable on 1/19/2008 *)
2996 (* interval verification in part4.cc:424186517+1 *)
2999 [((#4.0), x1, (square (#2.12)));
3000 ((#4.0), x2, square_2t0);
3001 ((#4.0), x3, square_2t0);
3002 ((#8.0), x4, (square (#3.2)));
3004 ((#4.0), x5, square_2t0);
3005 ((#4.0), x6, square_2t0)
3008 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.033))) \/
3009 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.8)))`;;
3011 (* interval verification in part4.cc:424186517+2 *)
3014 [(square_2t0, x1, (square (#2.696)));
3015 ((#4.0), x2, square_2t0);
3016 ((#4.0), x3, square_2t0);
3017 ((#8.0), x4, (square (#3.2)));
3019 ((#4.0), x5, square_2t0);
3020 ((#4.0), x6, square_2t0)
3023 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.058))) \/
3024 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.5)))`;;
3026 (* interval code in part4.cc:424186517+3 not used *)
3027 (* interval verification by Ferguson *)
3030 [(square_2t0, x1, (square (#2.696)));
3031 ((#4.0), x2, square_2t0);
3032 ((#4.0), x3, square_2t0);
3033 ((#8.0), x4, (square (#3.2)));
3035 ((#4.0), x5, square_2t0);
3036 ((#4.0), x6, square_2t0)
3039 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.073))) \/
3040 ( (eta_x x1 x2 x6) <. (sqrt (#2.0))))`;;
3046 LOC: 2002 k.c page 51
3047 17.25 Group_25 (pentagons)
3054 There are a few inequalities that arise for pentagonal regions.
3056 \begin{proposition} If the pentagonal region has no flat quarters
3057 and no upright quarters, the subregion $F$ is a pentagon. It
3069 \begin{proof} The proof is by deformations and interval calculations. If
3070 a deformation produces a new flat quarter, then the result follows
3071 from \cite[$\A_{13}$]{part4} and Inequality \ref {app:hexquad}. So
3072 we may assume that all diagonals remain at least $2\sqrt2$. If all
3073 diagonals remain at least 3.2, the result follows from the
3074 tcc-bound on the pentagon \cite[Section 5.5]{part4}. Thus, we
3075 assume that some diagonal is at most $3.2$. We deform the cluster
3077 $$(a_1,2,a_2,2,a_3,2,a_4,2,a_5,2),\quad |v_i|=a_i\in\{2,2t_0\}.$$
3078 Assume that $|v_1-v_3|\le3.2$. If $\max(a_1,a_3)=2t_0$, the
3079 result follows from \cite[$\A_{13}$]{part4} and
3080 Section~\ref{app:hexquad}, Equations \ref{eqn:hexquadsig} and
3081 \ref{eqn:hexquadtau}.
3083 Assume $a_1=a_3=2$. There is a diagonal of the quadrilateral of
3084 length at most $3.23$ because
3085 $$\Delta(3.23^2,4,4,3.23^2,4,3.2^2)<0.$$
3086 The result now follows from the following interval arithmetic
3089 (These inequalities are closely related to
3090 \cite[$\A_{21}$]{part4}.)
3094 (* interval verification by Ferguson *)
3097 [((#8.0), x4, (square (#3.2)));
3098 ((#8.0), x4', (square (#3.23)))
3100 ( ( (vor_0_x (#4.0) (#4.0) (#4.0) x4 (#4.0) (#4.0)) +.
3101 (vor_0_x (#4.0) (#4.0) (#4.0) x4' (#4.0) (#4.0)) +.
3102 (vor_0_x (#4.0) (#4.0) (#4.0) x4 x4' (#4.0))) <. (--. (#0.128)))`;;
3106 (* interval verification by Ferguson *)
3109 [((#8.0), x4, (square (#3.2)));
3110 ((#8.0), x4', (square (#3.23)))
3112 ( ( (tau_0_x (#4.0) (#4.0) (#4.0) x4 (#4.0) (#4.0)) +.
3113 (tau_0_x (#4.0) (#4.0) (#4.0) x4' (#4.0) (#4.0)) +.
3114 (tau_0_x (#4.0) (#4.0) (#4.0) x4 x4' (#4.0))) >. (#0.36925))`;;
3119 (* interval verification (commented out) in partK.cc *)
3120 (* interval verification by Ferguson *)
3123 [((#4.0), x3, square_2t0);
3124 ((#8.0), x4, (square (#3.06)))
3126 ( (tau_0_x (#4.0) (#4.0) x3 x4 (#4.0) (#4.0)) <.
3127 (tau_0_x square_2t0 (#4.0) x3 x4 (#4.0) (#4.0)))`;;
3132 (* interval verification (commented out) in partK.cc *)
3133 (* interval verification by Ferguson *)
3136 [((#4.0), x3, square_2t0);
3137 ((#8.0), x4, (square (#3.06)))
3139 ( (vor_0_x (#4.0) (#4.0) x3 x4 (#4.0) (#4.0)) >.
3140 (vor_0_x square_2t0 (#4.0) x3 x4 (#4.0) (#4.0)))`;;
3146 (* interval verification in partK.cc *)
3149 [((#4.0), x3, square_2t0);
3150 ((square (#3.06)), x5, (square (#3.23)));
3151 ((square (#3.06)), x6, (square (#3.23)))
3153 ( (vor_0_x (#4.0) (#4.0) x3 (#4.0) x5 x6) <. (--. (#0.128)))`;;
3157 (* interval verification in partK.cc *)
3160 [((square (#3.06)), x5, (square (#3.23)));
3161 ((square (#3.06)), x6, (square (#3.23)))
3163 ( (tau_0_x (#4.0) (#4.0) (#4.0) (#4.0) x5 x6) >. (#0.36925))`;;
3168 (* interval verification in partK.cc *)
3171 [((square (#3.06)), x5, (square (#3.23)));
3172 ((square (#3.06)), x6, (square (#3.23)))
3174 ( (tau_0_x (#4.0) (#4.0) square_2t0 (#4.0) x5 x6) >. (#0.31))`;;
3181 [((#4.0), x2, square_2t0);
3182 ((#4.0), x3, square_2t0);
3183 ((#8.0), x5, (square (#3.23)));
3185 ((square (#3.06)), x6, (square (#3.23)))
3187 ( (vor_0_x (#4.0) x2 x3 (#4.0) x5 x6) <.
3188 ( (--. (#0.137)) +. ( (--. (#0.14)) *. ( (sqrt x5) +.
3189 ( (--. (#2.0)) *. (sqrt (#2.0)))))))`;;
3193 (* interval verification in partK.cc *)
3196 [((#4.0), x2, square_2t0);
3197 ((#4.0), x3, square_2t0);
3198 ((#8.0), x5, (square (#3.23)));
3200 ((square (#3.105)), x6, (square (#3.23)))
3202 ( (tau_0_x (#4.0) x2 x3 (#4.0) x5 x6) >.
3203 ( (#0.31) +. ( (#0.14) *. ( (sqrt x5) +.
3204 ( (--. (#2.0)) *. (sqrt (#2.0)))))))`;;
3209 (* interval verification in partK.cc *)
3212 [((#4.0), x2, square_2t0);
3213 ((#4.0), x3, square_2t0);
3214 ((#8.0), x5, (square (#3.23)));
3216 ((square (#3.06)), x6, (square (#3.105)))
3218 ( (tau_0_x (#4.0) x2 x3 (#4.0) x5 x6) >.
3219 ( (#0.31) +. ( (#0.14) *. ( (sqrt x5) +. ( (--. (#2.0)) *. (sqrt (#2.0))))) +.
3220 ( (#0.19) *. ( (--. (#3.105)) +. (sqrt x6)))))`;;
3224 (* interval verification in partK.cc *)
3227 [((#4.0), x3, square_2t0);
3228 ((#8.0), x5, (square (#3.23)))
3230 ( (vor_0_x (#4.0) (#4.0) x3 (#4.0) x5 (#4.0)) <.
3231 ( (#0.009) +. ( (#0.14) *. ( (sqrt x5) +. ( (--. (#2.0)) *. (sqrt (#2.0)))))))`;;
3235 (* I_775220784 has been deprecated *)
3237 (* interval verification in partK.cc *)
3240 [((#4.0), x1, square_2t0);
3241 ((#4.0), x2, square_2t0);
3242 ((#8.0), x4, (square (#3.23)))
3244 ( (tau_0_x x1 x2 square_2t0 x4 (#4.0) (#4.0)) >. (#0.05925))`;;
3248 (* interval verification in partK.cc *)
3251 [((square (#3.06)), x4, (square (#3.105)))
3253 ( (tau_0_x square_2t0 (#4.0) (#4.0) x4 (#4.0) (#4.0)) >.
3254 ( (--. (#0.19)) *. ( (sqrt x4) +. (--. (#3.105)))))`;;
3259 LOC: 2002 k.c page 52
3267 Let $Q$ be a quadrilateral region with parameters
3268 $$(a_1,2t_0,a_2,2,a_3,2,a_4,2t_0),\quad a_i\in\{2,2t_0\}.$$
3269 Assume that $|v_2-v_4|\in[2\sqrt2,3.2]$,
3270 $|v_1-v_3|\in[3.2,3.46]$. Note that
3271 $$\Delta(4,4,8,2t_0^2,2t_0^2,3.46^2)<0.$$
3273 Sat Feb 21 12:47:03 EST 2004: Are we making an implicit convexity
3274 assumption by phrasing the inequality this way?
3279 (* interval verification by Ferguson *)
3280 let I_302085207_GEN=
3281 `\ a1 a2 a3 a4. (ineq
3283 ((#8.0),x4,(square (#3.2)))
3285 ((vor_0_x a1 a2 a4 x4 (square_2t0) (square_2t0) +
3286 (vor_0_x a3 a2 a4 x4 (#4.0) (#4.0)) <. (--. (#0.168))) \/
3287 ((cross_diag_x a1 a2 a4 x4 (square_2t0) (square_2t0) a3 (#4.0) (#4.0)) <. (#3.2)) \/
3288 ((cross_diag_x a1 a2 a4 x4 (square_2t0) (square_2t0) a3 (#4.0) (#4.0)) >. (#3.46))))`;;
3290 (* interval verification by Ferguson *)
3292 all_forall (list_mk_comb( I_302085207_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
3294 (* interval verification by Ferguson *)
3296 all_forall (list_mk_comb( I_302085207_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
3298 (* interval verification by Ferguson *)
3300 all_forall (list_mk_comb( I_302085207_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
3302 (* interval verification by Ferguson *)
3304 all_forall (list_mk_comb( I_302085207_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
3306 (* interval verification by Ferguson *)
3308 all_forall (list_mk_comb( I_302085207_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
3311 This seems unfeasible due to cross_diag constraints
3313 (* interval verification by Ferguson *)
3315 all_forall (list_mk_comb( I_302085207_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
3317 (* interval verification by Ferguson *)
3319 all_forall (list_mk_comb( I_302085207_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
3321 (* interval verification by Ferguson *)
3323 all_forall (list_mk_comb( I_302085207_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
3325 (* interval verification by Ferguson *)
3327 all_forall (list_mk_comb( I_302085207_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
3329 (* interval verification by Ferguson *)
3331 all_forall (list_mk_comb( I_302085207_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
3333 (* interval verification by Ferguson *)
3335 all_forall (list_mk_comb( I_302085207_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
3337 (* interval verification by Ferguson *)
3339 all_forall (list_mk_comb( I_302085207_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
3341 (* interval verification by Ferguson *)
3343 all_forall (list_mk_comb( I_302085207_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
3345 (* interval verification by Ferguson *)
3347 all_forall (list_mk_comb( I_302085207_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
3349 (* interval verification by Ferguson *)
3351 all_forall (list_mk_comb( I_302085207_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
3353 (* interval verification by Ferguson *)
3355 all_forall (list_mk_comb( I_302085207_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
3357 (* interval verification by Ferguson *)
3358 let I_411491283_GEN=
3359 `\ a1 a2 a3 a4. (ineq
3361 ((#8.0),x4,(square (#3.2)))
3363 ((tau_0_x a1 a2 a4 x4 (square_2t0) (square_2t0) +
3364 (tau_0_x a3 a2 a4 x4 (#4.0) (#4.0)) >. ( (#0.352))) \/
3365 ((cross_diag_x a1 a2 a4 x4 (square_2t0) (square_2t0) a3 (#4.0) (#4.0)) <. (#3.2)) \/
3366 ((cross_diag_x a1 a2 a4 x4 (square_2t0) (square_2t0) a3 (#4.0) (#4.0)) >. (#3.46))))`;;
3368 (* interval verification by Ferguson *)
3370 all_forall (list_mk_comb( I_411491283_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
3372 (* interval verification by Ferguson *)
3374 all_forall (list_mk_comb( I_411491283_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
3376 (* interval verification by Ferguson *)
3378 all_forall (list_mk_comb( I_411491283_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
3380 (* interval verification by Ferguson *)
3382 all_forall (list_mk_comb( I_411491283_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
3384 (* interval verification by Ferguson *)
3386 all_forall (list_mk_comb( I_411491283_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
3389 WWW Seems infeasible due to cross_diag_x constraints
3391 (* interval verification by Ferguson *)
3393 all_forall (list_mk_comb( I_411491283_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
3395 (* interval verification by Ferguson *)
3397 all_forall (list_mk_comb( I_411491283_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
3399 (* interval verification by Ferguson *)
3401 all_forall (list_mk_comb( I_411491283_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
3403 (* interval verification by Ferguson *)
3405 all_forall (list_mk_comb( I_411491283_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
3407 (* interval verification by Ferguson *)
3409 all_forall (list_mk_comb( I_411491283_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
3411 (* interval verification by Ferguson *)
3413 all_forall (list_mk_comb( I_411491283_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
3415 (* interval verification by Ferguson *)
3417 all_forall (list_mk_comb( I_411491283_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
3419 (* interval verification by Ferguson *)
3421 all_forall (list_mk_comb( I_411491283_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
3423 (* interval verification by Ferguson *)
3425 all_forall (list_mk_comb( I_411491283_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
3427 (* interval verification by Ferguson *)
3429 all_forall (list_mk_comb( I_411491283_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
3431 (* interval verification by Ferguson *)
3433 all_forall (list_mk_comb( I_411491283_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
3440 LOC: 2002 k.c page 52
3446 Consider a pentagonal region. If the pentagonal region has one
3447 flat quarter and no upright quarters, there is a quadrilateral
3448 region $F$. It satisfies
3456 Break the cluster into two simplices $S=S(y_1,\ldots,y_6)$,
3457 $S'=S(y'_1,y_2,y_3,y_4,y'_5,y'_6)$, by drawing a diagonal of
3458 length $y_4$. Assume that the edge $y'_5\in[2t_0,2\sqrt2]$. Let
3459 $y_4'$ be the length of the diagonal that crosses $y_4$.
3462 \vor_0 &< 2.1327-0.1 y_1 -0.15 y_2 -0.08 y_3 -0.15 y_5\\
3463 &\qquad -0.15 y_6 - 0.1 y'_1 - 0.17 y'_5 -0.16 y'_6,\\
3464 &\quad\text{if }\dih(S)<1.9,\ \dih(S')<2.0,\ y_1\in[2,2.2],\
3466 \vor_0 & < 2.02644 - 0.1 y_1 -0.14 (y_2+y_3)-0.15 (y_5+y_6)
3467 -0.1 y'_1 - 0.12 (y_5'+y_6'), \\
3468 &\quad\text{if }y_1\in[2,2.08],\quad y_4\le3.\\
3469 \vor_0 &+0.419351 \sol < 0.4542 + 0.0238 (y_5+y_6+y_6'),\\
3470 &\quad\text{if }\ y_4,y_4'\ge2\sqrt2.\\
3475 The inequalities above are verified in smaller pieces:
3481 (* interval verification in partK.cc *)
3482 (* CCC added delta >= 0 *)
3485 [((#4.0), x1, (square (#2.2)));
3486 ((#4.0), x2, square_2t0);
3487 ((#4.0), x3, square_2t0);
3488 ((#8.0), x4, square_4t0);
3489 ((#4.0), x5, square_2t0);
3490 ((#4.0), x6, square_2t0)
3493 ( (vor_0_x x1 x2 x3 x4 x5 x6) <.
3494 ( (#1.01) +. ( (--. (#0.1)) *. (sqrt x1)) +. ( (--. (#0.05)) *. (sqrt x2)) +. ( (--. (#0.05)) *. (sqrt x3)) +.
3495 ( (--. (#0.15)) *. (sqrt x5)) +. ( (--. (#0.15)) *. (sqrt x6)))) \/
3496 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.9)) \/
3497 ((delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
3501 (* interval verification in partK.cc *)
3504 [((#4.0), x1, square_2t0);
3505 ((#4.0), x2, square_2t0);
3506 ((#4.0), x3, square_2t0);
3507 ((#8.0), x4, square_4t0);
3509 (square_2t0, x5, (#8.0));
3510 ((#4.0), x6, square_2t0)
3513 ( (vor_0_x x1 x2 x3 x4 x5 x6) <.
3514 ( (#1.1227) +. ( (--. (#0.1)) *. (sqrt x1)) +. ( (--. (#0.1)) *. (sqrt x2)) +. ( (--. (#0.03)) *. (sqrt x3)) +.
3515 ( (--. (#0.17)) *. (sqrt x5)) +. ( (--. (#0.16)) *. (sqrt x6)))) \/
3516 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.0)) \/
3517 ( ( (sqrt x2) +. (sqrt x3)) >. (#4.67)))`;;
3521 (* interval verification in partK.cc *)
3524 [((#4.0), x1, (square (#2.08)));
3525 ((#4.0), x2, square_2t0);
3526 ((#4.0), x3, square_2t0);
3527 ((#8.0), x4, (#9.0));
3529 ((#4.0), x5, square_2t0);
3530 ((#4.0), x6, square_2t0)
3532 ( (vor_0_x x1 x2 x3 x4 x5 x6) <.
3533 ( (#1.0159) +. ( (--. (#0.1)) *. (sqrt x1)) +. ( (--. (#0.08)) *. ( (sqrt x2) +. (sqrt x3))) +.
3534 ( (#0.04) *. (sqrt x4)) +. ( (--. (#0.15)) *. ( (sqrt x5) +. (sqrt x6)))))`;;
3538 (* interval verification in partK.cc *)
3541 [((#4.0), x1, square_2t0);
3542 ((#4.0), x2, square_2t0);
3543 ((#4.0), x3, square_2t0);
3544 ((#8.0), x4, (#9.0));
3546 (square_2t0, x5, (#8.0));
3547 ((#4.0), x6, square_2t0)
3549 ( (vor_0_x x1 x2 x3 x4 x5 x6) <.
3550 ( (#1.01054) +. ( (--. (#0.1)) *. (sqrt x1)) +. ( (--. (#0.06)) *. ( (sqrt x2) +. (sqrt x3))) +.
3551 ( (--. (#0.04)) *. (sqrt x4)) +. ( (--. (#0.12)) *. ( (sqrt x5) +. (sqrt x6)))))`;;
3556 (* interval verification in partK.cc *)
3557 (* CCC i think you need delta constraints, added. *)
3560 [((#4.0), x1, square_2t0);
3561 ((#4.0), x2, square_2t0);
3562 ((#4.0), x3, square_2t0);
3563 ((#8.0), x4, square_4t0);
3565 ((#4.0), x5, square_2t0);
3566 ((#4.0), x6, square_2t0);
3567 ((#4.0), x1', square_2t0);
3568 (square_2t0, x5', (#8.0));
3570 ((#4.0), x6', square_2t0)
3573 ( ( (vor_0_x x1 x2 x3 x4 x5 x6) +. (vor_0_x x1' x2 x3 x4 x5' x6') +.
3574 ( (#0.419351) *. ( (sol_x x1 x2 x3 x4 x5 x6) +. (sol_x x1' x2 x3 x4 x5' x6')))) <.
3575 ( (#0.4542) +. ( (#0.0238) *. ( (sqrt x5) +. (sqrt x6) +. (sqrt x6'))))) \/
3576 ( (cross_diag_x x1 x2 x3 x4 x5 x6 x1' x5' x6') <. ( (#2.0) *. (sqrt (#2.0)))) \/
3577 ((delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/
3578 ((delta_x x1' x2 x3 x4 x5' x6') <. (#0.0)))`;;
3582 LOC: 2002 k.c page 53
3591 \vor_0 < -0.136\quad\text{and }\tau_0 > 0.224,
3595 for a combination of anchored simplex $S$ and special simplex
3596 $S'$, with $y_1(S)\in[2.696,2\sqrt2]$,
3597 $y_2(S),y_6(S)\in[2.45,2t_0]$, $y_4(S)\in[2\sqrt2,3.2]$, and with
3598 cross-diagonal at least $2t_0$. This inequality can be verified by
3599 proving the following inequalities in lower dimension. In the
3600 first four $y_1\in[2.696,2\sqrt2]$, $y_2,y_6\in[2.45,2t_0]$,
3601 $y_4\in[2\sqrt2,3.2]$, and $y_4'\ge2t_0$ (the cross-diagonal).
3607 (* interval verification by Ferguson *)
3610 [((square (#2.696)), x1, (#8.0));
3611 ((square (#2.45)), x2, square_2t0);
3612 ((#4.0), x3, square_2t0);
3613 ((#8.0), x4, (square (#3.2)));
3615 ((#4.0), x5, square_2t0);
3616 ((square (#2.45)), x6, square_2t0)
3619 ( ( (vor_0_x x1 x2 x3 x4 x5 x6) +. (vor_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))) <. (--. (#0.136))) \/
3620 ( (cross_diag_x x1 x2 x3 x4 x5 x6 (#4.0) (#4.0) (#4.0) ) <. two_t0))`;;
3625 (* interval verification by Ferguson *)
3628 [((square (#2.696)), x1, (#8.0));
3629 ((square (#2.45)), x2, square_2t0);
3630 ((#4.0), x3, square_2t0);
3631 ((#8.0), x4, (square (#3.2)));
3633 ((#4.0), x5, square_2t0);
3634 ((square (#2.45)), x6, square_2t0)
3637 ( ( (vor_0_x x1 x2 x3 x4 x5 x6) +. (vor_0_x square_2t0 x2 x3 x4 (#4.0) (#4.0))) <. (--. (#0.136))) \/
3638 ( (cross_diag_x x1 x2 x3 x4 x5 x6 square_2t0 (#4.0) (#4.0) ) <. two_t0))`;;
3643 (* interval verification by Ferguson *)
3646 [((square (#2.696)), x1, (#8.0));
3647 ((square (#2.45)), x2, square_2t0);
3648 ((#4.0), x3, square_2t0);
3649 ((#8.0), x4, (square (#3.2)));
3651 ((#4.0), x5, square_2t0);
3652 ((square (#2.45)), x6, square_2t0)
3655 ( ( (tau_0_x x1 x2 x3 x4 x5 x6) +.
3656 (tau_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))) >. (#0.224)) \/
3657 ( (cross_diag_x x1 x2 x3 x4 x5 x6 (#4.0) (#4.0) (#4.0) ) <. two_t0))`;;
3662 (* interval verification by Ferguson *)
3665 [((square (#2.696)), x1, (#8.0));
3666 ((square (#2.45)), x2, square_2t0);
3667 ((#4.0), x3, square_2t0);
3668 ((#8.0), x4, (square (#3.2)));
3670 ((#4.0), x5, square_2t0);
3671 ((square (#2.45)), x6, square_2t0)
3674 ( ( (tau_0_x x1 x2 x3 x4 x5 x6) +.
3675 (tau_0_x square_2t0 x2 x3 x4 (#4.0) (#4.0))) >. (#0.224)) \/
3676 ( (cross_diag_x x1 x2 x3 x4 x5 x6 square_2t0 (#4.0) (#4.0) ) <. two_t0))`;;
3681 (* interval verification by Ferguson *)
3684 [((square (#2.696)), x1, (#8.0));
3685 ((square (#2.45)), x2, square_2t0);
3686 ((#4.0), x3, square_2t0);
3687 ((#4.0), x4, square_2t0);
3689 (square_2t0, x5, square_2t0);
3690 ((square (#2.45)), x6, square_2t0)
3692 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.125)))`;;
3696 (* interval verification by Ferguson *)
3699 [((square (#2.696)), x1, (#8.0));
3700 ((#4.0), x2, square_2t0);
3701 ((#4.0), x3, square_2t0);
3702 ((#4.0), x4, square_2t0);
3704 (square_2t0, x5, square_2t0);
3705 ((#4.0), x6, square_2t0)
3707 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (#0.011))`;;
3712 (* interval verification by Ferguson *)
3715 [((square (#2.696)), x1, (#8.0));
3716 ((square (#2.45)), x2, square_2t0);
3717 ((#4.0), x3, square_2t0);
3718 ((#4.0), x4, square_2t0);
3720 (square_2t0, x5, square_2t0);
3721 ((square (#2.45)), x6, square_2t0)
3723 ( (tau_0_x x1 x2 x3 x4 x5 x6) >. (#0.17))`;;
3727 (* interval verification by Ferguson *)
3730 [((square (#2.696)), x1, (#8.0));
3731 ((#4.0), x2, square_2t0);
3732 ((#4.0), x3, square_2t0);
3733 ((#4.0), x4, square_2t0);
3735 (square_2t0, x5, square_2t0);
3736 ((#4.0), x6, square_2t0)
3738 ( (tau_0_x x1 x2 x3 x4 x5 x6) >. (#0.054))`;;
3743 LOC: 2002 k.c page 53
3751 $$\vor_0 < -0.24\text{ and }\tau_0 > 0.346,
3755 for an anchored simplex $S$ and simplex $S'$ with edge parameters
3756 $(3,2)$ in a hexagonal cluster, with $y_2(S)=y_2(S')$,
3757 $y_3(S)=y_3(S')$, $y_4(S)=y_4(S')$, $y_1(S)\in[2.696,2\sqrt2]$,
3758 $y_4(S)\in[2\sqrt2,3.2]$, $y_2(S),y_6(S)\in[2.45,2t_0]$, and
3759 $$\max(y_5(S'),y_6(S'))\in[2t_0,2\sqrt2],\quad
3760 \min(y_5(S'),y_6(S'))\in[2,2t_0].$$ This breaks into separate
3761 interval calculations for $S$ and $S'$.
3763 This inequality results from the following four inequalities:
3765 (* interval verification by Ferguson *)
3766 $\vor_0(S) < -0.126$ and $\tau_0(S) > 0.16$ \refno{369386367\dag}
3768 $\vor_0(S') < -0.114$ and $\tau_0(S') >0.186$ (There are two cases
3769 for each, depending on which of $y_5,y_6$ is longer.)
3770 (* interval verification by Ferguson *)
3771 \refno{724943459\dag}
3773 Sun Feb 22 07:47:31 EST 2004: I assume S' is a special below.
3780 [((square (#2.696)), x1, (#8.0));
3781 ((square (#2.45)), x2, square_2t0);
3782 ((#4.0), x3, square_2t0);
3783 ((#8.0), x4, (square (#3.2)));
3784 ((#4.0), x5, square_2t0);
3785 ((square (#2.45)), x6, square_2t0)
3787 (vor_0_x x1 x2 x3 x4 x5 x6 <. (--. (#0.126)))
3792 [((square (#2.696)), x1, (#8.0));
3793 ((square (#2.45)), x2, square_2t0);
3794 ((#4.0), x3, square_2t0);
3795 ((#8.0), x4, (square (#3.2)));
3796 ((#4.0), x5, square_2t0);
3797 ((square (#2.45)), x6, square_2t0)
3799 (tau_0_x x1 x2 x3 x4 x5 x6 >. (#0.16))
3804 [((#4.0), x1, square_2t0);
3805 ((square (#2.45)), x2, square_2t0);
3806 ((#4.0), x3, square_2t0);
3807 ((#8.0), x4, (square (#3.2)));
3808 ((#4.0), x5, square_2t0);
3809 ((square_2t0), x6, (#8.0))
3811 (vor_0_x x1 x2 x3 x4 x5 x6 <. (--. (#0.114)))
3816 [((#4.0), x1, square_2t0);
3817 ((square (#2.45)), x2, square_2t0);
3818 ((#4.0), x3, square_2t0);
3819 ((#8.0), x4, (square (#3.2)));
3820 ((square_2t0), x5, (#8.0));
3821 ((#4.0), x6, square_2t0)
3823 (vor_0_x x1 x2 x3 x4 x5 x6 <. (--. (#0.114)))
3828 [((#4.0), x1, square_2t0);
3829 ((square (#2.45)), x2, square_2t0);
3830 ((#4.0), x3, square_2t0);
3831 ((#8.0), x4, (square (#3.2)));
3832 ((#4.0), x5, square_2t0);
3833 ((square_2t0), x6, (#8.0))
3835 (tau_0_x x1 x2 x3 x4 x5 x6 >. (#0.186))
3840 [((#4.0), x1, square_2t0);
3841 ((square (#2.45)), x2, square_2t0);
3842 ((#4.0), x3, square_2t0);
3843 ((#8.0), x4, (square (#3.2)));
3844 ((square_2t0), x5, (#8.0));
3845 ((#4.0), x6, square_2t0)
3847 (tau_0_x x1 x2 x3 x4 x5 x6 >. (#0.186))
3852 LOC: 2002 k.c page 53
3856 (* interval verification by Ferguson *)
3857 (* CCC delta constraints added *)
3860 [((#4.0), x1, square_2t0);
3861 ((#4.0), x2, square_2t0);
3862 ((square(#2.45)), x3, square_2t0);
3863 ((#8.0), x4, square_4t0);
3864 (square_2t0, x5, (#8.0));
3865 ((#4.0), x6, square_2t0);
3866 ((#4.0), x7, square_2t0);
3867 ((#4.0), x8, square_2t0);
3868 ((#4.0), x9, square_2t0)]
3869 (((vor_0_x x1 x2 x3 x4 x5 x6) +
3870 (vor_0_x x7 x2 x3 x4 x8 x9) <. (-- (#0.149)))
3872 (cross_diag_x x1 x2 x3 x4 x5 x6 x7 x8 x9 <. (sqrt8)) \/
3873 ((delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/
3874 ((delta_x x7 x2 x3 x4 x8 x9) <. (#0.0)))`;;
3876 (* CCC delta constraints added *)
3879 [((#4.0), x1, square_2t0);
3880 ((#4.0), x2, square_2t0);
3881 ((square(#2.45)), x3, square_2t0);
3882 ((#8.0), x4, square_4t0);
3883 (square_2t0, x5, (#8.0));
3884 ((#4.0), x6, square_2t0);
3885 ((#4.0), x7, square_2t0);
3886 ((#4.0), x8, square_2t0);
3887 ((#4.0), x9, square_2t0)]
3888 (((tau_0_x x1 x2 x3 x4 x5 x6) +
3889 (tau_0_x x7 x2 x3 x4 x8 x9) >. (#0.281))
3891 (cross_diag_x x1 x2 x3 x4 x5 x6 x7 x8 x9 <. (sqrt8)) \/
3892 ((delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/
3893 ((delta_x x7 x2 x3 x4 x8 x9) <. (#0.0)))`;;
3898 LOC: 2002 k.c page 54
3904 $$\vor_0 < -0.254\quad\text{and }\tau_0 > 0.42625,
3907 %\oldlabel{A.4.12.9}
3908 for a combination of anchored simplex $S$ and quadrilateral
3909 cluster $Q$. It is assumed that $y_1(S)\in[2.696,2\sqrt2]$,
3910 $y_2(S),y_6(S)\in[2.45,2t_0]$. The adjacent quadrilateral
3911 subcluster is assumed to have both diagonals $\ge2\sqrt2$, and
3913 $$(a_1,b_1,a_2,b_2,a_3,b_3,a_4,b_4),$$
3914 with $b_4\in[2\sqrt2,3.2]$. The verification of this inequality
3915 reduces to separate inequalities for the anchored simplex and
3916 quadrilateral subcluster. For the anchored simplex we use the
3917 bounds $\vor_0(S')<-0.126$, $\tau_0(S')>0.16$ that have already
3918 been established above. We then show that the quad cluster
3921 (* interval verification by Ferguson *)
3922 $\vor_0 < -0.128$ and $\tau_0 > 0.26625$. \refno{327474205\dag}
3924 (* interval verification in partK.cc *)
3925 For this, use deformations to reduce either to the case where the
3926 diagonal is $2\sqrt2$, or to the case where $b_1=b_2=b_3=2$,
3927 $a_2,a_3\in\{2,2t_0\}$. When the diagonal is $2\sqrt2$, the flat
3928 quarter can be scored by \cite[$\A_{13}$]{part4}:
3929 $(\vor_0<0.009,\tau_0>0.05925)$.
3930 (There are two cases depending on which direction the diagonal of
3931 length $2\sqrt2$ runs.)
3937 (* CCC delta constraints added *)
3938 (* XXX fixed syntax *)
3948 ((square(#2.45)), x1, square_2t0);
3949 ((#4.0), x3, square_2t0);
3950 ((#8.0), x4, square_4t0);
3951 ((#8.0), x5, (square (#3.2)));
3952 ((#4.0), x6, square_2t0)]
3953 (((vor_0_x x1 x2 x3 x4 x5 x6) +
3954 (vor_0_x x7 x2 x3 x4 x8 x9) <. (-- (#0.128)))
3956 (cross_diag_x x1 x2 x3 x4 x5 x6 x7 x8 x9 <. (sqrt8)) \/
3957 ((delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/
3958 ((delta_x x7 x2 x3 x4 x8 x9) <. (#0.0)))`;;
3960 (* XXX fixed syntax *)
3961 (* CCC delta constraints added *)
3964 let x2 = (square_2t0) in
3970 ((square(#2.45)), x1, square_2t0);
3971 ((#4.0), x3, square_2t0);
3972 ((#8.0), x4, square_4t0);
3973 ((#8.0), x5, (square (#3.2)));
3974 ((#4.0), x6, square_2t0)]
3975 (((vor_0_x x1 x2 x3 x4 x5 x6) +
3976 (vor_0_x x7 x2 x3 x4 x8 x9) <. (-- (#0.128)))
3978 (cross_diag_x x1 x2 x3 x4 x5 x6 x7 x8 x9 <. (sqrt8)) \/
3979 ((delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/
3980 ((delta_x x7 x2 x3 x4 x8 x9) <. (#0.0)))`;;
3982 (* XXX fixed syntax *)
3985 let x2 = (square_2t0) in
3986 let x7 = (square_2t0) in
3991 ((square(#2.45)), x1, square_2t0);
3992 ((#4.0), x3, square_2t0);
3993 ((#8.0), x4, square_4t0);
3994 ((#8.0), x5, (square (#3.2)));
3995 ((#4.0), x6, square_2t0)]
3996 (((vor_0_x x1 x2 x3 x4 x5 x6) +
3997 (vor_0_x x7 x2 x3 x4 x8 x9) <. (-- (#0.128)))
3999 (cross_diag_x x1 x2 x3 x4 x5 x6 x7 x8 x9
4002 (* XXX fixed syntax *)
4006 let x7 = (square_2t0) in
4011 ((square(#2.45)), x1, square_2t0);
4012 ((#4.0), x3, square_2t0);
4013 ((#8.0), x4, square_4t0);
4014 ((#8.0), x5, (square (#3.2)));
4015 ((#4.0), x6, square_2t0)]
4016 (((vor_0_x x1 x2 x3 x4 x5 x6) +
4017 (vor_0_x x7 x2 x3 x4 x8 x9) <. (-- (#0.128)))
4019 (cross_diag_x x1 x2 x3 x4 x5 x6 x7 x8 x9
4025 ((square(#2.45)), x1, square_2t0);
4026 ((#4.0), x2, square_2t0);
4027 ((#4.0), x3, square_2t0);
4028 ((#8.0), x5, (square (#3.2)));
4029 ((#4.0), x6, square_2t0)]
4030 ((vor_0_x x1 x2 x3 (#8.0) x5 x6) <. (-- (#0.128)) - (#0.009))`;;
4035 ((square(#2.45)), x1, square_2t0);
4036 ((#4.0), x2, square_2t0);
4037 ((#4.0), x3, square_2t0);
4038 ((#4.0), x4, square_2t0);
4039 ((#8.0), x5, (square (#3.2)))]
4040 ((vor_0_x x1 x2 x3 x4 x5 (#8.0)) <. (-- (#0.128)) - (#0.009))`;;
4044 CCC delta constraints added here as well.
4056 ((square(#2.45)), x1, square_2t0);
4057 ((#4.0), x3, square_2t0);
4058 ((#8.0), x4, square_4t0);
4059 ((#8.0), x5, (square (#3.2)));
4060 ((#4.0), x6, square_2t0)]
4061 ((((tau_0_x x1 x2 x3 x4 x5 x6) +
4062 (tau_0_x x7 x2 x3 x4 x8 x9) >. (#0.26625))
4064 (cross_diag_x x1 x2 x3 x4 x5 x6 x7 x8 x9 <. (sqrt8))) \/
4065 ((delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/
4066 ((delta_x x7 x2 x3 x4 x8 x9) <. (#0.0)))`;;
4070 let x2 = (square_2t0) in
4076 ((square(#2.45)), x1, square_2t0);
4077 ((#4.0), x3, square_2t0);
4078 ((#8.0), x4, square_4t0);
4079 ((#8.0), x5, (square (#3.2)));
4080 ((#4.0), x6, square_2t0)]
4081 ((((tau_0_x x1 x2 x3 x4 x5 x6) +
4082 (tau_0_x x7 x2 x3 x4 x8 x9) >. (#0.26625))
4084 (cross_diag_x x1 x2 x3 x4 x5 x6 x7 x8 x9 <. (sqrt8))) \/
4085 ((delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/
4086 ((delta_x x7 x2 x3 x4 x8 x9) <. (#0.0)))`;;
4090 let x2 = (square_2t0) in
4091 let x7 = (square_2t0) in
4096 ((square(#2.45)), x1, square_2t0);
4097 ((#4.0), x3, square_2t0);
4098 ((#8.0), x4, square_4t0);
4099 ((#8.0), x5, (square (#3.2)));
4100 ((#4.0), x6, square_2t0)]
4101 (((tau_0_x x1 x2 x3 x4 x5 x6) +
4102 (tau_0_x x7 x2 x3 x4 x8 x9) >. (#0.26625))
4104 (cross_diag_x x1 x2 x3 x4 x5 x6 x7 x8 x9
4110 let x7 = (square_2t0) in
4115 ((square(#2.45)), x1, square_2t0);
4116 ((#4.0), x3, square_2t0);
4117 ((#8.0), x4, square_4t0);
4118 ((#8.0), x5, (square (#3.2)));
4119 ((#4.0), x6, square_2t0)]
4120 (((tau_0_x x1 x2 x3 x4 x5 x6) +
4121 (tau_0_x x7 x2 x3 x4 x8 x9) >. (#0.26625))
4123 (cross_diag_x x1 x2 x3 x4 x5 x6 x7 x8 x9
4129 ((square(#2.45)), x1, square_2t0);
4130 ((#4.0), x2, square_2t0);
4131 ((#4.0), x3, square_2t0);
4132 ((#8.0), x5, (square (#3.2)));
4133 ((#4.0), x6, square_2t0)]
4134 ((tau_0_x x1 x2 x3 (#8.0) x5 x6) >. (#0.26625) - (#0.05925))`;;
4139 ((square(#2.45)), x1, square_2t0);
4140 ((#4.0), x2, square_2t0);
4141 ((#4.0), x3, square_2t0);
4142 ((#4.0), x4, square_2t0);
4143 ((#8.0), x5, (square (#3.2)))]
4144 ((tau_0_x x1 x2 x3 x4 x5 (#8.0)) >. (#0.26625) - (#0.05925))`;;
4148 LOC: 2002 k.c page 55--
4149 18. Appendix Hexagonal Inequalities
4153 LOC: 2002 k.c page 55--56
4154 SKIP 18.1. This has been moved to the main part of the
4155 text. It is more mathematical argument than interval arithmetic.
4159 LOC: 2002 k.c page 56--59
4160 SKIP first part of 18.2.
4161 This is a mathematical proof. It has been moved into the main
4166 LOC: 2002 k.c page 56--59
4171 (* interval verification by Ferguson *)
4174 [((#4.0), x1, square_2t0);
4175 ((#4.0), x2, square_2t0);
4176 ((#4.0), x3, square_2t0);
4177 ((square (#3.2)), x4, (square (#3.2)));
4179 ((#8.0), x5, (square (#3.2)));
4180 ((#4.0), x6, square_2t0)
4182 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.212)) +. (--. (#0.0461)) +. (#0.137)))`;;
4186 (* interval verification by Ferguson *)
4189 [((#4.0), x1, square_2t0);
4190 ((#4.0), x2, square_2t0);
4191 ((#4.0), x3, square_2t0);
4192 ((square (#3.2)), x4, (square (#3.2)));
4194 ((#8.0), x5, (square (#3.2)));
4195 ((#4.0), x6, square_2t0)
4197 ( (tau_0_x x1 x2 x3 x4 x5 x6) >. ( (#0.54525) +. (--. (#0.0)) +. (--. (#0.31))))`;;
4202 LOC: 2002 k.c page 59
4208 let CKC_377409251= (* 18.3 : app:p1b *)
4209 let CKC_586214007= (* 18.4 : app:p1c *)
4210 let CKC_89384104= (* 18.5 : app:p1d *)
4211 let CKC_859726639= (* kc group 18.6 : app:p1e *)
4212 let CKC_673399623= (* kc group 18.7 : app:p2a *)
4213 let CKC_297256991= (* kc group 18.8 : app:p2b *)
4214 let CKC_861511432= (* kc group 18.9 : app:p2c *)
4215 let CKC_746445726= (* kc group 18.10 : app:p2d *)
4216 let CKC_897046482= (* kc group 18.11 : app:p2e *)
4217 let CKC_928952883= (* kc group 18.12 : app:p2f *)
4218 let CKC_673800906= (* kc group 18.13 : app:p2g *)
4219 let CKC_315678695= (* kc group 18.14 : app:p3 *)
4220 let CKC_468742136= (* kc group 18.15 : app:p8 *)
4221 let CKC_938091791= (* kc group 18.16 : app:p11 *)
4225 (* interval verification by Ferguson *)
4228 (* This old code is incorrect.
4229 let I_583626763_GEN=
4233 ((square(#3.2)), x, (#16.0));
4234 ((square(#3.2)), x', square_4t0)
4236 (((vor_0_x (#4.0) a2 a3 (#4.0) x (#4.0))+.
4237 (vor_0_x (#4.0) a3 a4 (#4.0) x' x) +.
4238 (vor_0_x (#4.0) a4 (#4.0) (#4.0) (#8.0) x') + (#0.0461)
4241 (delta_x (#4.0) a2 a3 (#4.0) x (#4.0) <. (#0.0)) \/
4242 (delta_x (#4.0) a3 a4 (#4.0) x' x <. (#0.0)) \/
4243 (delta_x (#4.0) a4 (#4.0) (#4.0) (#8.0) x'<. (#0.0)))))`;;
4246 (* The diagonals of the pentagon
4247 run between (v1,v3) and (v3,v5). The long edge
4248 of the pentagon is (v1,v5). See SPVI,2002,page 60,group 18.3. *)
4250 let I_583626763_GEN=
4254 ((square(#3.2)), x, (#16.0));
4255 ((square(#3.2)), x', square_4t0)
4257 (((vor_0_x (#4.0) a2 a3 (#4.0) x (#4.0))+.
4258 (vor_0_x a3 (#4.0) (#4.0) (#8.0) x x') +.
4259 (vor_0_x (#4.0) a4 a3 (#4.0) x' (#4.0)) + (#0.0461)
4262 (delta_x (#4.0) a2 a3 (#4.0) x (#4.0) <. (#0.0)) \/
4263 (delta_x a3 (#4.0) (#4.0) (#8.0) x x' <. (#0.0)) \/
4264 (delta_x (#4.0) a4 a3 (#4.0) x' (#4.0) <. (#0.0)))))`;;
4266 (* interval verification by Ferguson *)
4268 (* False for old code
4270 Bound: 0.189116321203
4271 Point: [10.2399999999, 14.9282032302]
4276 (list_mk_comb(I_583626763_GEN,[`#4.0`;`#4.0`;`#4.0`]));;
4279 (* interval verification by Ferguson *)
4281 (* False for old code
4282 Bound: 0.265976192226
4284 Point: [10.2399999999, 18.1174102784]
4288 (list_mk_comb(I_583626763_GEN,[`#4.0`;`#4.0`;`square_2t0`]));;
4291 (* interval verification by Ferguson *)
4292 (* False for old code
4293 Bound: 0.626837612707
4295 Point: [11.8474915071, 14.9282032302]
4299 (list_mk_comb(I_583626763_GEN,[`#4.0`;`square_2t0`;`#4.0`]));;
4302 (* interval verification by Ferguson *)
4303 (* False for old code
4305 Bound: 0.607887643248
4307 Point: [11.0313746566, 18.1174102783]
4311 (list_mk_comb(I_583626763_GEN,[`#4.0`;`square_2t0`;`square_2t0`]));;
4314 (* interval verification by Ferguson *)
4315 (* WWW Infeasible old code *)
4318 (list_mk_comb(I_583626763_GEN,[`square_2t0`;`#4.0`;`#4.0`]));;
4321 (* interval verification by Ferguson *)
4322 (* WWW Infeasible old code *)
4325 (list_mk_comb(I_583626763_GEN,[`square_2t0`;`#4.0`;`square_2t0`]));;
4328 (* interval verification by Ferguson *)
4329 (* False for old code *)
4332 (list_mk_comb(I_583626763_GEN,[`square_2t0`;`square_2t0`;`#4.0`]));;
4335 (* interval verification by Ferguson *)
4336 (* False for old code *)
4339 (list_mk_comb(I_583626763_GEN,[`square_2t0`;`square_2t0`;`square_2t0`]));;
4342 (* All false or infeasible for old code. This had the same
4343 diagonals error as 583626763. See comments there. *)
4344 (* interval verification by Ferguson *)
4347 let I_390951718_GEN=
4351 ((square(#3.2)), x, (#16.0));
4352 ((square(#3.2)), x', square_4t0)
4354 (((tau_0_x (#4.0) a2 a3 (#4.0) x (#4.0))+.
4355 (tau_0_x (#4.0) a3 a4 (#4.0) x' x) +.
4356 (tau_0_x (#4.0) a4 (#4.0) (#4.0) (#8.0) x')
4359 (delta_x (#4.0) a2 a3 (#4.0) x (#4.0) <. (#0.0)) \/
4360 (delta_x (#4.0) a3 a4 (#4.0) x' x <. (#0.0)) \/
4361 (delta_x (#4.0) a4 (#4.0) (#4.0) (#8.0) x'<. (#0.0)))))`;;
4367 (* The diagonals of the pentagon
4368 run between (v1,v3) and (v3,v5). The long edge
4369 of the pentagon is (v1,v5). See SPVI,2002,page 60,group 18.3. *)
4371 let I_390951718_GEN =
4375 ((square(#3.2)), x, (#16.0));
4376 ((square(#3.2)), x', square_4t0)
4378 (((tau_0_x (#4.0) a2 a3 (#4.0) x (#4.0))+.
4379 (tau_0_x a3 (#4.0) (#4.0) (#8.0) x x') +.
4380 (tau_0_x (#4.0) a4 a3 (#4.0) x' (#4.0))
4383 (delta_x (#4.0) a2 a3 (#4.0) x (#4.0) <. (#0.0)) \/
4384 (delta_x a3 (#4.0) (#4.0) (#8.0) x x' <. (#0.0)) \/
4385 (delta_x (#4.0) a4 a3 (#4.0) x' (#4.0) <. (#0.0)))))`;;
4387 (* interval verification by Ferguson *)
4390 (list_mk_comb(I_390951718_GEN,[`#4.0`;`#4.0`;`#4.0`]));;
4392 (* interval verification by Ferguson *)
4395 (list_mk_comb(I_390951718_GEN,[`#4.0`;`#4.0`;`square_2t0`]));;
4397 (* interval verification by Ferguson *)
4400 (list_mk_comb(I_390951718_GEN,[`#4.0`;`square_2t0`;`#4.0`]));;
4402 (* interval verification by Ferguson *)
4405 (list_mk_comb(I_390951718_GEN,[`#4.0`;`square_2t0`;`square_2t0`]));;
4407 (* interval verification by Ferguson *)
4410 (list_mk_comb(I_390951718_GEN,[`square_2t0`;`#4.0`;`#4.0`]));;
4412 (* interval verification by Ferguson *)
4415 (list_mk_comb(I_390951718_GEN,[`square_2t0`;`#4.0`;`square_2t0`]));;
4417 (* interval verification by Ferguson *)
4420 (list_mk_comb(I_390951718_GEN,[`square_2t0`;`square_2t0`;`#4.0`]));;
4422 (* interval verification by Ferguson *)
4425 (list_mk_comb(I_390951718_GEN,[`square_2t0`;`square_2t0`;`square_2t0`]));;
4427 let CKC_377409251= (* 18.3 *)
4429 [I_390951718_8; I_390951718_7; I_390951718_6; I_390951718_5;
4430 I_390951718_4; I_390951718_3; I_390951718_2; I_390951718_1;
4431 I_583626763_8; I_583626763_7; I_583626763_6; I_583626763_5;
4432 I_583626763_4; I_583626763_3; I_583626763_2; I_583626763_1; ];;
4435 LOC: 2002 k.c page 59
4439 (* interval verification by Ferguson *)
4441 (* added disjunct on 3/11/2008 to express that |v2-v4|\ge sqrt8.
4442 This is not in the statement of 2002. Note added to DCG errata.
4443 This does not affect the proof, because this conditions holds in
4444 practice. I haven't traced the error in the original code.
4445 It is quite possible that Ferguson inserts this condition and
4446 it never got updated in the text. *)
4448 let I_621852152_GEN=
4452 ((#8.0),b5,(square (#3.2)));
4453 ((square(#3.2)), x, (square_4t0));
4454 ((square(#3.2)), x', (square_4t0))
4456 (((vor_0_x a3 a2 a1 (#4.0) x (#4.0)) +.
4457 (vor_0_x a3 a1 a5 b5 x' x) +.
4458 (vor_0_x a3 a5 a4 (#4.0) (#4.0) x')
4459 + (#0.0461) <. (--(#0.212)))
4461 (dih_x a3 a2 a1 (#4.0) x (#4.0) +.
4462 dih_x a3 a1 a5 b5 xp x +.
4463 dih_x a3 a5 a4 (#4.0) (#4.0) xp <. dih_x a3 a2 a4 (#8.0) (#4.0) (#4.0)) \/
4464 (delta_x a3 a2 a1 (#4.0) x (#4.0) <. (#0.0)) \/
4465 (delta_x a3 a1 a5 b5 x' x <. (#0.0)) \/
4466 (delta_x a3 a5 a4 (#4.0) (#4.0) x' <. (#0.0)))))`;;
4468 (* interval verification by Ferguson *)
4471 (list_mk_comb(I_621852152_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
4473 (* interval verification by Ferguson *)
4476 (list_mk_comb(I_621852152_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
4479 (* interval verification by Ferguson *)
4482 (list_mk_comb(I_621852152_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
4484 (* interval verification by Ferguson *)
4487 (list_mk_comb(I_621852152_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
4489 (* interval verification by Ferguson *)
4492 (list_mk_comb(I_621852152_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
4494 (* interval verification by Ferguson *)
4496 (* CCC false , disjunct added
4497 Bound: 0.0571539662754
4499 Point: [8.00048294466, 13.920039161, 15.2775848381]
4504 (list_mk_comb(I_621852152_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
4506 (* interval verification by Ferguson *)
4509 (list_mk_comb(I_621852152_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
4511 (* interval verification by Ferguson *)
4514 (list_mk_comb(I_621852152_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
4516 (* WWW infeasible *)
4517 (* interval verification by Ferguson *)
4520 (list_mk_comb(I_621852152_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
4522 (* WWW infeasible *)
4523 (* interval verification by Ferguson *)
4526 (list_mk_comb(I_621852152_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
4528 (* WWW infeasible *)
4529 (* interval verification by Ferguson *)
4532 (list_mk_comb(I_621852152_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
4534 (* WWW infeasible *)
4535 (* interval verification by Ferguson *)
4538 (list_mk_comb(I_621852152_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
4540 (* interval verification by Ferguson *)
4543 (list_mk_comb(I_621852152_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
4545 (* interval verification by Ferguson *)
4548 (list_mk_comb(I_621852152_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
4550 (* interval verification by Ferguson *)
4553 (list_mk_comb(I_621852152_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
4555 (* interval verification by Ferguson *)
4558 (list_mk_comb(I_621852152_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
4560 (* interval verification by Ferguson *)
4563 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
4565 (* interval verification by Ferguson *)
4567 CCC false, disjunct added
4568 Bound: 0.0270250652729
4570 Point: [8.00060070939, 13.9200391262, 13.920039283]
4574 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
4576 (* interval verification by Ferguson *)
4579 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
4581 (* interval verification by Ferguson *)
4584 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
4586 (* interval verification by Ferguson *)
4588 CCC false, disjunct added
4589 Bound: 0.0571539734352
4591 Point: [8.00048294461, 15.2775848381, 13.920039161]
4596 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
4598 (* interval verification by Ferguson *)
4600 CCC false, disjunct added
4601 Bound: 0.0813970415878
4603 Point: [8.00208732876, 15.2775848587, 15.2775793515]
4608 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
4610 (* interval verification by Ferguson *)
4613 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
4615 (* interval verification by Ferguson *)
4618 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
4620 (* interval verification by Ferguson *)
4623 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
4625 (* interval verification by Ferguson *)
4628 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
4630 (* WWW infeasible *)
4631 (* interval verification by Ferguson *)
4634 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
4636 (* interval verification by Ferguson *)
4639 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
4641 (* interval verification by Ferguson *)
4644 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
4646 (* interval verification by Ferguson *)
4649 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
4651 (* interval verification by Ferguson *)
4654 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
4656 (* interval verification by Ferguson *)
4659 (list_mk_comb(I_621852152_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
4663 (* interval verification by Ferguson, source/section_a46_1c.c *)
4664 (* There are counterexamples to various cases, listed below as stated
4665 in 2002. The version below inserts an extra dihedral constraint
4666 that is satisfied in practice. *)
4668 let I_207203174_GEN=
4672 ((#8.0),b5,(square (#3.2)));
4673 ((square(#3.2)), x, (square_4t0));
4674 ((square(#3.2)), x', (square_4t0))
4676 (((tau_0_x a3 a2 a1 (#4.0) x (#4.0)) +.
4677 (tau_0_x a3 a1 a5 b5 x' x) +.
4678 (tau_0_x a3 a5 a4 (#4.0) (#4.0) x')
4680 (dih_x a3 a2 a1 (#4.0) x (#4.0) +.
4681 dih_x a3 a1 a5 b5 xp x +.
4682 dih_x a3 a5 a4 (#4.0) (#4.0) xp <. dih_x a3 a2 a4 (#8.0) (#4.0) (#4.0)) \/
4683 (delta_x a3 a2 a1 (#4.0) x (#4.0) <. (#0.0)) \/
4684 (delta_x a3 a2 a1 (#4.0) x (#4.0) <. (#0.0)) \/
4685 (delta_x a3 a1 a5 b5 x' x <. (#0.0)) \/
4686 (delta_x a3 a5 a4 (#4.0) (#4.0) x' <. (#0.0)))))`;;
4688 (* interval verification by Ferguson *)
4691 (list_mk_comb(I_207203174_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
4693 (* interval verification by Ferguson *)
4696 (list_mk_comb(I_207203174_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
4698 (* interval verification by Ferguson *)
4699 (* WWW infeasible *)
4702 (list_mk_comb(I_207203174_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
4704 (* interval verification by Ferguson *)
4707 (list_mk_comb(I_207203174_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
4709 (* interval verification by Ferguson *)
4712 (list_mk_comb(I_207203174_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
4714 (* interval verification by Ferguson *)
4715 (* CCC false , extra constraint added
4716 False at {ay1,ay2,ay3,ay4,ay5}={2,2,2.51,2,2.51};
4717 by5=Sqrt[8]; {y,yp}={3.2,3.9086};
4718 Note that Solve[Delta[2, 2.51, 2.51, y, 2, 2] == 0, y] gives a zero
4720 tauVc drops rapidly as x' increases in the range [3.9,3.9086].
4721 It is still true by a considerable margin at yp=3.9.
4723 The verification code is there in source/section_a46_1c.c, but I haven't
4726 Reported in dcg_errata.tex 1/31/2008, TCH.
4730 (list_mk_comb(I_207203174_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
4732 (* interval verification by Ferguson *)
4735 (list_mk_comb(I_207203174_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
4737 (* interval verification by Ferguson *)
4740 (list_mk_comb(I_207203174_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
4743 (* interval verification by Ferguson *)
4744 (* WWW infeasible *)
4747 (list_mk_comb(I_207203174_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
4749 (* interval verification by Ferguson *)
4750 (* WWW infeasible *)
4753 (list_mk_comb(I_207203174_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
4755 (* interval verification by Ferguson *)
4756 (* WWW infeasible *)
4759 (list_mk_comb(I_207203174_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
4761 (* interval verification by Ferguson *)
4762 (* WWW infeasible *)
4765 (list_mk_comb(I_207203174_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
4767 (* interval verification by Ferguson *)
4770 (list_mk_comb(I_207203174_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
4772 (* interval verification by Ferguson *)
4775 (list_mk_comb(I_207203174_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
4777 (* interval verification by Ferguson *)
4780 (list_mk_comb(I_207203174_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
4782 (* interval verification by Ferguson *)
4785 (list_mk_comb(I_207203174_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
4787 (* interval verification by Ferguson *)
4790 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
4792 (* interval verification by Ferguson *)
4795 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
4797 (* interval verification by Ferguson *)
4798 (* WWW infeasible *)
4801 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
4803 (* interval verification by Ferguson *)
4806 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
4808 (* interval verification by Ferguson *)
4809 (* CCC false. extra constraint added. Comments before I_207203174_6 *)
4812 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
4814 (* interval verification by Ferguson *)
4815 (* CCC false. extra constraint added. Comments before I_207203174_6 *)
4818 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
4820 (* interval verification by Ferguson *)
4823 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
4825 (* interval verification by Ferguson *)
4828 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
4830 (* interval verification by Ferguson *)
4833 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
4835 (* interval verification by Ferguson *)
4838 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
4840 (* interval verification by Ferguson *)
4841 (* WWW infeasible *)
4844 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
4846 (* interval verification by Ferguson *)
4849 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
4851 (* interval verification by Ferguson *)
4854 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
4856 (* interval verification by Ferguson *)
4859 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
4861 (* interval verification by Ferguson *)
4864 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
4866 (* interval verification by Ferguson *)
4869 (list_mk_comb(I_207203174_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
4872 let CKC_586214007= (* 18.4 *)
4874 I_207203174_32;I_207203174_31;I_207203174_30;I_207203174_29;
4875 I_207203174_28;I_207203174_27;I_207203174_26;I_207203174_25;
4876 I_207203174_24;I_207203174_23;I_207203174_22;I_207203174_21;
4877 I_207203174_20;I_207203174_19;I_207203174_18;I_207203174_17;
4878 I_207203174_16;I_207203174_15;I_207203174_14;I_207203174_13;
4879 I_207203174_12;I_207203174_11;I_207203174_10;I_207203174_9;
4880 I_207203174_8;I_207203174_7;I_207203174_6;I_207203174_5;
4881 I_207203174_4;I_207203174_3;I_207203174_2;I_207203174_1;
4882 I_621852152_32;I_621852152_31;I_621852152_30;I_621852152_29;
4883 I_621852152_28;I_621852152_27;I_621852152_26;I_621852152_25;
4884 I_621852152_24;I_621852152_23;I_621852152_22;I_621852152_21;
4885 I_621852152_20;I_621852152_19;I_621852152_18;I_621852152_17;
4886 I_621852152_16;I_621852152_15;I_621852152_14;I_621852152_13;
4887 I_621852152_12;I_621852152_11;I_621852152_10;I_621852152_9;
4888 I_621852152_8;I_621852152_7;I_621852152_6;I_621852152_5;
4889 I_621852152_4;I_621852152_3;I_621852152_2;I_621852152_1; ];;
4893 LOC: 2002 k.c page 59
4898 (* interval verification by Ferguson *)
4899 let I_368258024_GEN=
4900 `(\ a1 a2 a3 a4 a5 a6.
4903 ((#8.0) , xd3, (square(#3.2)));
4904 ((square(#3.2)), xd4 , square_4t0);
4905 ((#8.0) , xd5,(square(#3.2)))
4907 (((vor_0_x a1 a2 a3 (#4.0) xd3 (#4.0)) +.
4908 (vor_0_x a1 a3 a4 (#4.0) xd4 xd3) +.
4909 (vor_0_x a1 a4 a5 (#4.0) xd5 xd4) +.
4910 (vor_0_x a1 a5 a6 (#4.0) (#4.0) xd5)
4913 (cross_diag_x a3 a1 a4 xd4 (#4.0) xd3 a5 (#4.0) xd5
4915 (delta_x a1 a2 a3 (#4.0) xd3 (#4.0) <. (#0.0)) \/
4916 (delta_x a1 a3 a4 (#4.0) xd4 xd3 <. (#0.0)) \/
4917 (delta_x a1 a4 a5 (#4.0) xd5 xd4 <. (#0.0)) \/
4918 (delta_x a1 a5 a6 (#4.0) (#4.0) xd5 <. (#0.0)))))`;;
4920 (* interval verification by Ferguson *)
4923 Bound: 0.894112044825
4925 Point: [8.27682664562, 15.0624674033, 8.27682846171]
4927 Fixed. The sign on the cross-diag inequalty was reversed.
4930 {y3, y4, y5} = Sqrt[{8.27682664562, 15.0624674033, 8.27682846171}];
4931 Enclosed[2, 2, y3, 2, y4, 2, 2, 2, y5]
4932 This yields 0.00216981, but the cross_diag_x constraint should keep
4938 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
4940 (* interval verification by Ferguson *)
4941 (* CCC See comments on _1 *)
4944 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
4946 (* interval verification by Ferguson *)
4949 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
4951 (* interval verification by Ferguson *)
4954 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
4956 (* interval verification by Ferguson *)
4959 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
4961 (* interval verification by Ferguson *)
4964 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
4966 (* interval verification by Ferguson *)
4969 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
4971 (* interval verification by Ferguson *)
4974 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
4977 (* interval verification by Ferguson *)
4980 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
4982 (* interval verification by Ferguson *)
4985 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
4987 (* interval verification by Ferguson *)
4990 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
4992 (* interval verification by Ferguson *)
4995 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
4997 (* interval verification by Ferguson *)
5000 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
5002 (* interval verification by Ferguson *)
5005 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5007 (* interval verification by Ferguson *)
5010 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5012 (* interval verification by Ferguson *)
5015 (list_mk_comb(I_368258024_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5017 (* interval verification by Ferguson *)
5020 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
5022 (* interval verification by Ferguson *)
5025 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
5027 (* interval verification by Ferguson *)
5030 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
5032 (* interval verification by Ferguson *)
5035 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5037 (* interval verification by Ferguson *)
5040 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
5042 (* interval verification by Ferguson *)
5045 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5047 (* interval verification by Ferguson *)
5050 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5052 (* interval verification by Ferguson *)
5055 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5057 (* interval verification by Ferguson *)
5060 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
5062 (* interval verification by Ferguson *)
5065 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
5067 (* interval verification by Ferguson *)
5070 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
5072 (* interval verification by Ferguson *)
5075 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5077 (* interval verification by Ferguson *)
5080 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
5082 (* interval verification by Ferguson *)
5085 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5087 (* interval verification by Ferguson *)
5090 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5092 (* interval verification by Ferguson *)
5095 (list_mk_comb(I_368258024_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5097 (* interval verification by Ferguson *)
5100 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
5102 (* interval verification by Ferguson *)
5105 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
5107 (* interval verification by Ferguson *)
5110 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
5112 (* interval verification by Ferguson *)
5115 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5117 (* interval verification by Ferguson *)
5120 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
5122 (* interval verification by Ferguson *)
5125 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5127 (* interval verification by Ferguson *)
5130 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5132 (* interval verification by Ferguson *)
5135 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5138 (* interval verification by Ferguson *)
5141 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
5143 (* interval verification by Ferguson *)
5146 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
5148 (* interval verification by Ferguson *)
5151 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
5153 (* interval verification by Ferguson *)
5156 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5158 (* interval verification by Ferguson *)
5161 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
5163 (* interval verification by Ferguson *)
5166 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5168 (* interval verification by Ferguson *)
5171 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5173 (* interval verification by Ferguson *)
5176 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5178 (* interval verification by Ferguson *)
5181 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
5183 (* interval verification by Ferguson *)
5186 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
5188 (* interval verification by Ferguson *)
5191 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
5193 (* interval verification by Ferguson *)
5196 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5198 (* interval verification by Ferguson *)
5201 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
5203 (* interval verification by Ferguson *)
5206 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5208 (* interval verification by Ferguson *)
5211 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5213 (* interval verification by Ferguson *)
5216 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5218 (* interval verification by Ferguson *)
5221 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
5223 (* interval verification by Ferguson *)
5226 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
5228 (* interval verification by Ferguson *)
5231 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
5233 (* interval verification by Ferguson *)
5236 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5238 (* interval verification by Ferguson *)
5241 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
5243 (* interval verification by Ferguson *)
5246 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5248 (* interval verification by Ferguson *)
5251 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5253 (* interval verification by Ferguson *)
5256 (list_mk_comb(I_368258024_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5259 (* interval verification by Ferguson *)
5260 (* CCC all fail/infeasible. Fixed cross-diag sign. Apply comments from 368258024. *)
5261 let I_564618342_GEN=
5262 `(\ a1 a2 a3 a4 a5 a6.
5265 ((#8.0) , xd3, (square(#3.2)));
5266 ((square(#3.2)), xd4 , square_4t0);
5267 ((#8.0) , xd5,(square(#3.2)))
5269 (((tau_0_x a1 a2 a3 (#4.0) xd3 (#4.0)) +.
5270 (tau_0_x a1 a3 a4 (#4.0) xd4 xd3) +.
5271 (tau_0_x a1 a4 a5 (#4.0) xd5 xd4) +.
5272 (tau_0_x a1 a5 a6 (#4.0) (#4.0) xd5)
5275 (cross_diag_x a3 a1 a4 xd4 (#4.0) xd3 a5 (#4.0) xd5
5277 (delta_x a1 a2 a3 (#4.0) xd3 (#4.0) <. (#0.0)) \/
5278 (delta_x a1 a3 a4 (#4.0) xd4 xd3 <. (#0.0)) \/
5279 (delta_x a1 a4 a5 (#4.0) xd5 xd4 <. (#0.0)) \/
5280 (delta_x a1 a5 a6 (#4.0) (#4.0) xd5 <. (#0.0)))))`;;
5282 (* interval verification by Ferguson *)
5285 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
5287 (* interval verification by Ferguson *)
5290 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
5292 (* interval verification by Ferguson *)
5295 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
5297 (* interval verification by Ferguson *)
5300 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5302 (* interval verification by Ferguson *)
5305 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
5307 (* interval verification by Ferguson *)
5310 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5312 (* interval verification by Ferguson *)
5315 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5317 (* interval verification by Ferguson *)
5320 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5323 (* interval verification by Ferguson *)
5326 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
5328 (* interval verification by Ferguson *)
5331 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
5333 (* interval verification by Ferguson *)
5336 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
5338 (* interval verification by Ferguson *)
5341 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5343 (* interval verification by Ferguson *)
5346 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
5348 (* interval verification by Ferguson *)
5351 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5353 (* interval verification by Ferguson *)
5356 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5358 (* interval verification by Ferguson *)
5361 (list_mk_comb(I_564618342_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5363 (* interval verification by Ferguson *)
5366 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
5368 (* interval verification by Ferguson *)
5371 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
5373 (* interval verification by Ferguson *)
5376 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
5378 (* interval verification by Ferguson *)
5381 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5383 (* interval verification by Ferguson *)
5386 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
5388 (* interval verification by Ferguson *)
5391 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5393 (* interval verification by Ferguson *)
5396 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5398 (* interval verification by Ferguson *)
5401 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5403 (* interval verification by Ferguson *)
5406 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
5408 (* interval verification by Ferguson *)
5411 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
5413 (* interval verification by Ferguson *)
5416 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
5418 (* interval verification by Ferguson *)
5421 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5423 (* interval verification by Ferguson *)
5426 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
5428 (* interval verification by Ferguson *)
5431 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5433 (* interval verification by Ferguson *)
5436 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5438 (* interval verification by Ferguson *)
5441 (list_mk_comb(I_564618342_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5443 (* interval verification by Ferguson *)
5446 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
5448 (* interval verification by Ferguson *)
5451 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
5453 (* interval verification by Ferguson *)
5456 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
5458 (* interval verification by Ferguson *)
5461 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5463 (* interval verification by Ferguson *)
5466 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
5468 (* interval verification by Ferguson *)
5471 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5473 (* interval verification by Ferguson *)
5476 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5478 (* interval verification by Ferguson *)
5481 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5484 (* interval verification by Ferguson *)
5487 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
5489 (* interval verification by Ferguson *)
5492 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
5494 (* interval verification by Ferguson *)
5497 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
5499 (* interval verification by Ferguson *)
5502 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5504 (* interval verification by Ferguson *)
5507 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
5509 (* interval verification by Ferguson *)
5512 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5514 (* interval verification by Ferguson *)
5517 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5519 (* interval verification by Ferguson *)
5522 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5524 (* interval verification by Ferguson *)
5527 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
5529 (* interval verification by Ferguson *)
5532 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
5534 (* interval verification by Ferguson *)
5537 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
5539 (* interval verification by Ferguson *)
5542 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5544 (* interval verification by Ferguson *)
5547 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
5549 (* interval verification by Ferguson *)
5552 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5554 (* interval verification by Ferguson *)
5557 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5559 (* interval verification by Ferguson *)
5562 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5564 (* interval verification by Ferguson *)
5567 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
5569 (* interval verification by Ferguson *)
5572 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
5574 (* interval verification by Ferguson *)
5577 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
5579 (* interval verification by Ferguson *)
5582 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5584 (* interval verification by Ferguson *)
5587 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
5589 (* interval verification by Ferguson *)
5592 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5594 (* interval verification by Ferguson *)
5597 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5599 (* interval verification by Ferguson *)
5602 (list_mk_comb(I_564618342_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5604 let CKC_89384104= (* 18.5 *)
5606 I_564618342_64;I_564618342_63;I_564618342_62;I_564618342_61;
5607 I_564618342_60;I_564618342_59;I_564618342_58;I_564618342_57;
5608 I_564618342_56;I_564618342_55;I_564618342_54;I_564618342_53;
5609 I_564618342_52;I_564618342_51;I_564618342_50;I_564618342_49;
5610 I_564618342_48;I_564618342_47;I_564618342_46;I_564618342_45;
5611 I_564618342_44;I_564618342_43;I_564618342_42;I_564618342_41;
5612 I_564618342_40;I_564618342_39;I_564618342_38;I_564618342_37;
5613 I_564618342_36;I_564618342_35;I_564618342_34;I_564618342_33;
5614 I_564618342_32;I_564618342_31;I_564618342_30;I_564618342_29;I_564618342_28;I_564618342_27;
5615 I_564618342_26;I_564618342_25;I_564618342_24;I_564618342_23;I_564618342_22;I_564618342_21;
5616 I_564618342_20;I_564618342_19;I_564618342_18;I_564618342_17;I_564618342_16;I_564618342_15;
5617 I_564618342_14;I_564618342_13;I_564618342_12;I_564618342_11;I_564618342_10;I_564618342_9;
5618 I_564618342_8;I_564618342_7;I_564618342_6;I_564618342_5;I_564618342_4;I_564618342_3;I_564618342_2;I_564618342_1;
5619 I_368258024_64;I_368258024_63;I_368258024_62;I_368258024_61;I_368258024_60;I_368258024_59;
5620 I_368258024_58;I_368258024_57;I_368258024_56;I_368258024_55;I_368258024_54;I_368258024_53;
5621 I_368258024_52;I_368258024_51;I_368258024_50;I_368258024_49;I_368258024_48;I_368258024_47;I_368258024_46;
5622 I_368258024_45;I_368258024_44;I_368258024_43;I_368258024_42;I_368258024_41;I_368258024_40;I_368258024_39;
5623 I_368258024_38;I_368258024_37;I_368258024_36;I_368258024_35;I_368258024_34;I_368258024_33;I_368258024_32;
5624 I_368258024_31;I_368258024_30;I_368258024_29;I_368258024_28;I_368258024_27;I_368258024_26;I_368258024_25;
5625 I_368258024_24;I_368258024_23;I_368258024_22;I_368258024_21;I_368258024_20;I_368258024_19;I_368258024_18;
5626 I_368258024_17;I_368258024_16;I_368258024_15;I_368258024_14;I_368258024_13;I_368258024_12;I_368258024_11;
5627 I_368258024_10;I_368258024_9;I_368258024_8;I_368258024_7;I_368258024_6;I_368258024_5;I_368258024_4;
5628 I_368258024_3;I_368258024_2;I_368258024_1; ];;
5632 LOC: 2002 k.c page 59
5637 (* interval verification by Ferguson *)
5638 (* CCC many fail/infeasible, cross diag constraint fixed. *)
5639 let I_498774382_GEN=
5640 `(\ a1 a2 a3 a4 a5 a6.
5643 ((#8.0) , x, (square(#3.2)));
5644 ((#8.0) , x'', (square(#3.2)));
5645 ((square(#3.2)), x' , (square(#3.78)))
5647 (((vor_0_x a1 a2 a6 x (#4.0) (#4.0) ) +
5648 (vor_0_x a2 a6 a5 (#4.0) x' x) +
5649 (vor_0_x a2 a3 a5 x'' x' (#4.0) ) +
5650 (vor_0_x a3 a4 a5 (#4.0) x'' (#4.0) )
5653 (cross_diag_x a3 a2 a5 x' x'' (#4.0) a6 (#4.0) x
5655 (delta_x a1 a2 a6 x (#4.0) (#4.0) <. (#0.0)) \/
5656 (delta_x a2 a6 a5 (#4.0) x' x <. (#0.0)) \/
5657 (delta_x a2 a3 a5 x'' x' (#4.0) <. (#0.0)) \/
5658 (delta_x a3 a4 a5 (#4.0) x'' (#4.0) <. (#0.0)))))`;;
5660 (* interval verification by Ferguson *)
5663 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
5665 (* interval verification by Ferguson *)
5668 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
5670 (* interval verification by Ferguson *)
5673 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
5675 (* interval verification by Ferguson *)
5678 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5680 (* interval verification by Ferguson *)
5683 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
5685 (* interval verification by Ferguson *)
5688 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5690 (* interval verification by Ferguson *)
5693 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5695 (* interval verification by Ferguson *)
5698 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5701 (* interval verification by Ferguson *)
5704 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
5706 (* interval verification by Ferguson *)
5709 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
5711 (* interval verification by Ferguson *)
5714 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
5716 (* interval verification by Ferguson *)
5719 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5721 (* interval verification by Ferguson *)
5724 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
5726 (* interval verification by Ferguson *)
5729 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5731 (* interval verification by Ferguson *)
5734 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5736 (* interval verification by Ferguson *)
5739 (list_mk_comb(I_498774382_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5741 (* interval verification by Ferguson *)
5744 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
5746 (* interval verification by Ferguson *)
5749 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
5751 (* interval verification by Ferguson *)
5754 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
5756 (* interval verification by Ferguson *)
5759 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5761 (* interval verification by Ferguson *)
5764 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
5766 (* interval verification by Ferguson *)
5769 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5771 (* interval verification by Ferguson *)
5774 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5776 (* interval verification by Ferguson *)
5779 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5781 (* interval verification by Ferguson *)
5784 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
5786 (* interval verification by Ferguson *)
5789 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
5791 (* interval verification by Ferguson *)
5794 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
5796 (* interval verification by Ferguson *)
5799 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5801 (* interval verification by Ferguson *)
5804 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
5806 (* interval verification by Ferguson *)
5809 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5811 (* interval verification by Ferguson *)
5814 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5816 (* interval verification by Ferguson *)
5819 (list_mk_comb(I_498774382_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5821 (* interval verification by Ferguson *)
5824 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
5826 (* interval verification by Ferguson *)
5829 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
5831 (* interval verification by Ferguson *)
5834 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
5836 (* interval verification by Ferguson *)
5839 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5841 (* interval verification by Ferguson *)
5844 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
5846 (* interval verification by Ferguson *)
5849 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5851 (* interval verification by Ferguson *)
5854 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5856 (* interval verification by Ferguson *)
5859 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5862 (* interval verification by Ferguson *)
5865 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
5867 (* interval verification by Ferguson *)
5870 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
5872 (* interval verification by Ferguson *)
5875 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
5877 (* interval verification by Ferguson *)
5880 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5882 (* interval verification by Ferguson *)
5885 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
5887 (* interval verification by Ferguson *)
5890 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5892 (* interval verification by Ferguson *)
5895 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5897 (* interval verification by Ferguson *)
5900 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5902 (* interval verification by Ferguson *)
5905 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
5907 (* interval verification by Ferguson *)
5910 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
5912 (* interval verification by Ferguson *)
5915 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
5917 (* interval verification by Ferguson *)
5920 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5922 (* interval verification by Ferguson *)
5925 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
5927 (* interval verification by Ferguson *)
5930 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5932 (* interval verification by Ferguson *)
5935 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5937 (* interval verification by Ferguson *)
5940 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5942 (* interval verification by Ferguson *)
5945 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
5947 (* interval verification by Ferguson *)
5950 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
5952 (* interval verification by Ferguson *)
5955 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
5957 (* interval verification by Ferguson *)
5960 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
5962 (* interval verification by Ferguson *)
5965 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
5967 (* interval verification by Ferguson *)
5970 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
5972 (* interval verification by Ferguson *)
5975 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
5977 (* interval verification by Ferguson *)
5980 (list_mk_comb(I_498774382_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
5983 (* interval verification by Ferguson *)
5984 (* CCC many fail/infeasible, cross diag fixed. *)
5985 let I_544865225_GEN=
5986 `(\ a1 a2 a3 a4 a5 a6.
5989 ((#8.0) , x, (square(#3.2)));
5990 ((#8.0) , x'', (square(#3.2)));
5991 ((square(#3.2)), x' , (square(#3.78)))
5993 (((tau_0_x a1 a2 a6 x (#4.0) (#4.0) ) +
5994 (tau_0_x a2 a6 a5 (#4.0) x' x) +
5995 (tau_0_x a2 a3 a5 x'' x' (#4.0) ) +
5996 (tau_0_x a3 a4 a5 (#4.0) x'' (#4.0) )
5999 (cross_diag_x a3 a2 a5 x' x'' (#4.0) a6 (#4.0) x
6001 (delta_x a1 a2 a6 x (#4.0) (#4.0) <. (#0.0)) \/
6002 (delta_x a2 a6 a5 (#4.0) x' x <. (#0.0)) \/
6003 (delta_x a2 a3 a5 x'' x' (#4.0) <. (#0.0)) \/
6004 (delta_x a3 a4 a5 (#4.0) x'' (#4.0) <. (#0.0)))))`;;
6006 (* interval verification by Ferguson *)
6009 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
6011 (* interval verification by Ferguson *)
6014 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
6016 (* interval verification by Ferguson *)
6019 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
6021 (* interval verification by Ferguson *)
6024 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
6026 (* interval verification by Ferguson *)
6029 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
6031 (* interval verification by Ferguson *)
6034 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
6036 (* interval verification by Ferguson *)
6039 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
6041 (* interval verification by Ferguson *)
6044 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
6047 (* interval verification by Ferguson *)
6050 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
6052 (* interval verification by Ferguson *)
6055 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
6057 (* interval verification by Ferguson *)
6060 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
6062 (* interval verification by Ferguson *)
6065 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
6067 (* interval verification by Ferguson *)
6070 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
6072 (* interval verification by Ferguson *)
6075 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
6077 (* interval verification by Ferguson *)
6080 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
6082 (* interval verification by Ferguson *)
6085 (list_mk_comb(I_544865225_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
6087 (* interval verification by Ferguson *)
6090 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
6092 (* interval verification by Ferguson *)
6095 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
6097 (* interval verification by Ferguson *)
6100 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
6102 (* interval verification by Ferguson *)
6105 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
6107 (* interval verification by Ferguson *)
6110 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
6112 (* interval verification by Ferguson *)
6115 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
6117 (* interval verification by Ferguson *)
6120 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
6122 (* interval verification by Ferguson *)
6125 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
6127 (* interval verification by Ferguson *)
6130 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
6132 (* interval verification by Ferguson *)
6135 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
6137 (* interval verification by Ferguson *)
6140 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
6142 (* interval verification by Ferguson *)
6145 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
6147 (* interval verification by Ferguson *)
6150 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
6152 (* interval verification by Ferguson *)
6155 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
6157 (* interval verification by Ferguson *)
6160 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
6162 (* interval verification by Ferguson *)
6165 (list_mk_comb(I_544865225_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
6167 (* interval verification by Ferguson *)
6170 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
6172 (* interval verification by Ferguson *)
6175 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
6177 (* interval verification by Ferguson *)
6180 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
6182 (* interval verification by Ferguson *)
6185 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
6187 (* interval verification by Ferguson *)
6190 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
6192 (* interval verification by Ferguson *)
6195 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
6197 (* interval verification by Ferguson *)
6200 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
6202 (* interval verification by Ferguson *)
6205 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
6208 (* interval verification by Ferguson *)
6211 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
6213 (* interval verification by Ferguson *)
6216 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
6218 (* interval verification by Ferguson *)
6221 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
6223 (* interval verification by Ferguson *)
6226 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
6228 (* interval verification by Ferguson *)
6231 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
6233 (* interval verification by Ferguson *)
6236 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
6238 (* interval verification by Ferguson *)
6241 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
6243 (* interval verification by Ferguson *)
6246 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
6248 (* interval verification by Ferguson *)
6251 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
6253 (* interval verification by Ferguson *)
6256 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
6258 (* interval verification by Ferguson *)
6261 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
6263 (* interval verification by Ferguson *)
6266 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
6268 (* interval verification by Ferguson *)
6271 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
6273 (* interval verification by Ferguson *)
6276 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
6278 (* interval verification by Ferguson *)
6281 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
6283 (* interval verification by Ferguson *)
6286 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
6288 (* interval verification by Ferguson *)
6291 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
6293 (* interval verification by Ferguson *)
6296 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
6298 (* interval verification by Ferguson *)
6301 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
6303 (* interval verification by Ferguson *)
6306 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
6308 (* interval verification by Ferguson *)
6311 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
6313 (* interval verification by Ferguson *)
6316 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
6318 (* interval verification by Ferguson *)
6321 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
6323 (* interval verification by Ferguson *)
6326 (list_mk_comb(I_544865225_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
6328 let CKC_859726639= list_mk_conj
6329 [I_544865225_64;I_544865225_63;I_544865225_62;I_544865225_61;I_544865225_60;I_544865225_59;
6330 I_544865225_58;I_544865225_57;I_544865225_56;I_544865225_55;I_544865225_54;I_544865225_53;
6331 I_544865225_52;I_544865225_51;I_544865225_50;I_544865225_49;I_544865225_48;I_544865225_47;
6332 I_544865225_46;I_544865225_45;I_544865225_44;I_544865225_43;I_544865225_42;I_544865225_41;
6333 I_544865225_40;I_544865225_39;I_544865225_38;I_544865225_37;I_544865225_36;I_544865225_35;
6334 I_544865225_34;I_544865225_33;I_544865225_32;I_544865225_31;I_544865225_30;I_544865225_29;
6335 I_544865225_28;I_544865225_27;I_544865225_26;I_544865225_25;I_544865225_24;I_544865225_23;
6336 I_544865225_22;I_544865225_21;I_544865225_20;I_544865225_19;I_544865225_18;I_544865225_17;
6337 I_544865225_16;I_544865225_15;I_544865225_14;I_544865225_13;I_544865225_12;I_544865225_11;
6338 I_544865225_10;I_544865225_9;I_544865225_8;I_544865225_7;I_544865225_6;I_544865225_5;I_544865225_4;
6339 I_544865225_3;I_544865225_2;I_544865225_1; I_498774382_64;I_498774382_63;I_498774382_62;I_498774382_61;
6340 I_498774382_60;I_498774382_59;I_498774382_58;I_498774382_57;I_498774382_56;I_498774382_55;I_498774382_54;
6341 I_498774382_53;I_498774382_52;I_498774382_51;I_498774382_50;I_498774382_49;I_498774382_48;I_498774382_47;
6342 I_498774382_46;I_498774382_45;I_498774382_44;I_498774382_43;I_498774382_42;I_498774382_41;I_498774382_40;
6343 I_498774382_39;I_498774382_38;I_498774382_37;I_498774382_36;I_498774382_35;I_498774382_34;I_498774382_33;
6344 I_498774382_32;I_498774382_31;I_498774382_30;I_498774382_29;I_498774382_28;I_498774382_27;I_498774382_26;
6345 I_498774382_25;I_498774382_24;I_498774382_23;I_498774382_22;I_498774382_21;I_498774382_20;I_498774382_19;
6346 I_498774382_18;I_498774382_17;I_498774382_16;I_498774382_15;I_498774382_14;I_498774382_13;I_498774382_12;
6347 I_498774382_11;I_498774382_10;I_498774382_9;I_498774382_8;I_498774382_7;I_498774382_6;I_498774382_5;
6348 I_498774382_4;I_498774382_3;I_498774382_2;I_498774382_1; ];; (* kc group 18.6 *)
6352 LOC: 2002 k.c page 59
6357 (* interval verification by Ferguson *)
6360 [((#4.0), x1, square_2t0);
6361 ((#4.0), x2, square_2t0);
6362 ((#4.0), x3, square_2t0);
6363 ((#8.0) , x4, (#8.0));
6364 ((#8.0), x5, (square(#3.2)));
6365 ((square_2t0), x6, (#8.0))
6368 ( (vor_0_x x1 x2 x3 x4 x5 x6 ) <. (--(#0.221))-(&.2)*(#0.009)))` ;;
6371 (* interval verification by Ferguson *)
6375 Bound: 0.322153452432
6377 Point: [4, 4.16407792566, 4, 7.99999999999, 10.2399999999, 8]
6379 Sign of the inequality was reversed. Fixed 1/31/2008
6384 [((#4.0), x1, square_2t0);
6385 ((#4.0), x2, square_2t0);
6386 ((#4.0), x3, square_2t0);
6387 ((#8.0) , x4, (#8.0));
6388 ((#8.0), x5, (square(#3.2)));
6389 ((square_2t0), x6, (#8.0))
6392 ( (tau_0_x x1 x2 x3 x4 x5 x6 ) >. (#0.486)-(&.2)*(#0.05925)))`;;
6394 let CKC_673399623= list_mk_conj [I_791682321;I_234734606; ];; (* kc group 18.7 *)
6397 LOC: 2002 k.c page 59
6401 (* interval verification by Ferguson *)
6402 (* cross-diag constraint fixed 1/31/2008 *)
6403 let I_995351614_GEN=
6407 ((#4.0) , a1, square_2t0);
6408 ((#8.0) , x, (square(#3.2)));
6409 ((#8.0) , b1, (square(#3.2)))
6411 ((((vor_0_x a1 a2 a4 x square_2t0 b1) +
6412 (vor_0_x a3 a2 a4 x (#4.0) (#4.0) )
6413 <. (--(#0.221))-(#0.009)))
6415 (cross_diag_x a1 a2 a4 x square_2t0 b1 a3 (#4.0) (#4.0)
6417 (delta_x a1 a2 a4 x square_2t0 b1 <. (#0.0)) \/
6418 (delta_x a3 a2 a4 x (#4.0) (#4.0) <. (#0.0)))))`;;
6420 (* interval verification by Ferguson *)
6423 (list_mk_comb(I_995351614_GEN,[`#4.0`;`#4.0`;`#4.0`]));;
6425 (* interval verification by Ferguson *)
6428 (list_mk_comb(I_995351614_GEN,[`#4.0`;`#4.0`;`square_2t0`]));;
6430 (* interval verification by Ferguson *)
6433 (list_mk_comb(I_995351614_GEN,[`#4.0`;`square_2t0`;`#4.0`]));;
6435 (* interval verification by Ferguson *)
6438 (list_mk_comb(I_995351614_GEN,[`#4.0`;`square_2t0`;`square_2t0`]));;
6440 (* interval verification by Ferguson *)
6443 (list_mk_comb(I_995351614_GEN,[`square_2t0`;`#4.0`;`#4.0`]));;
6445 (* interval verification by Ferguson *)
6448 (list_mk_comb(I_995351614_GEN,[`square_2t0`;`#4.0`;`square_2t0`]));;
6450 (* interval verification by Ferguson *)
6453 (list_mk_comb(I_995351614_GEN,[`square_2t0`;`square_2t0`;`#4.0`]));;
6455 (* interval verification by Ferguson *)
6458 (list_mk_comb(I_995351614_GEN,[`square_2t0`;`square_2t0`;`square_2t0`]));;
6461 (* interval verification by Ferguson *)
6462 (* cross-diag constraint fixed 1/31/2008 *)
6464 let I_321843503_GEN=
6468 ((#4.0) , a1, square_2t0);
6469 ((#8.0) , x, (square(#3.2)));
6470 ((#8.0) , b1, (square(#3.2)))
6472 ((((tau_0_x a1 a2 a4 x square_2t0 b1) +
6473 (tau_0_x a3 a2 a4 x (#4.0) (#4.0) )
6474 >. (#0.486)-(#0.0595)))
6476 (cross_diag_x a1 a2 a4 x square_2t0 b1 a3 (#4.0) (#4.0)
6478 (delta_x a1 a2 a4 x square_2t0 b1 <. (#0.0)) \/
6479 (delta_x a3 a2 a4 x (#4.0) (#4.0) <. (#0.0)))))`;;
6481 (* interval verification by Ferguson *)
6484 (list_mk_comb(I_321843503_GEN,[`#4.0`;`#4.0`;`#4.0`]));;
6486 (* interval verification by Ferguson *)
6489 (list_mk_comb(I_321843503_GEN,[`#4.0`;`#4.0`;`square_2t0`]));;
6491 (* interval verification by Ferguson *)
6494 (list_mk_comb(I_321843503_GEN,[`#4.0`;`square_2t0`;`#4.0`]));;
6496 (* interval verification by Ferguson *)
6499 (list_mk_comb(I_321843503_GEN,[`#4.0`;`square_2t0`;`square_2t0`]));;
6501 (* interval verification by Ferguson *)
6504 (list_mk_comb(I_321843503_GEN,[`square_2t0`;`#4.0`;`#4.0`]));;
6506 (* interval verification by Ferguson *)
6509 (list_mk_comb(I_321843503_GEN,[`square_2t0`;`#4.0`;`square_2t0`]));;
6511 (* interval verification by Ferguson *)
6514 (list_mk_comb(I_321843503_GEN,[`square_2t0`;`square_2t0`;`#4.0`]));;
6516 (* interval verification by Ferguson *)
6519 (list_mk_comb(I_321843503_GEN,[`square_2t0`;`square_2t0`;`square_2t0`]));;
6521 let CKC_297256991= list_mk_conj [I_321843503_8;I_321843503_7;I_321843503_6;I_321843503_5;
6522 I_321843503_4;I_321843503_3;I_321843503_2;I_321843503_1; I_995351614_8;
6523 I_995351614_7;I_995351614_6;I_995351614_5;I_995351614_4;I_995351614_3;
6524 I_995351614_2;I_995351614_1; ];; (* kc group 18.8 *)
6527 LOC: 2002 k.c page 59--60
6531 (* interval verification by Ferguson, source/section_a46_2c.c *)
6534 Bound: 0.196433568955
6536 Point: [6.30009999999, 3.99999999999, 3.99999999999, 3.99999999999, 7.99999999999, 10.2399999999
6537 Typo: sqrt2 changed to sqrt8 below.
6538 The typo appears in SPVI2002,SPVI1998. Note added to dcg_errata 1/31/2008.
6545 [((#4.0), x1, square_2t0);
6546 ((#4.0), x2, square_2t0);
6547 ((#4.0), x3, square_2t0);
6548 ((#4.0) , x4, (#4.0));
6549 ((#8.0), x5, (square(#3.2)));
6550 ((square(#3.2)), x6, (square(#3.47)))
6553 ( (vor_0_x x1 x2 x3 x4 x5 x6 ) <. (--(#0.19))-((sqrt x5)-(sqrt8))*(#0.14)))`;;
6556 (* interval verification in partK.cc, possibly also in Ferguson *)
6559 [((#4.0), x1, square_2t0);
6560 ((#4.0), x2, square_2t0);
6561 ((#4.0), x3, square_2t0);
6562 ((#4.0) , x4, (#4.0));
6563 ((#8.0), x5, (square(#3.2)));
6564 ((square(#3.2)), x6, (square(#3.23)))
6567 ( (tau_0_x x1 x2 x3 x4 x5 x6 ) >. (#0.281)))`;;
6570 (* interval verification by Ferguson *)
6573 [((#4.0), x1, square_2t0);
6574 ((#4.0), x2, square_2t0);
6575 ((#4.0), x3, square_2t0);
6576 ((#4.0) , x4, (#4.0));
6577 (square_2t0, x5, square_2t0);
6578 ((square(#3.2)), x6, (square(#3.2)))
6581 ( (vor_0_x x1 x2 x3 x4 x5 x6 ) <. (--(#0.11))))`;;
6583 (* interval verification by Ferguson *)
6586 [((#4.0), x1, square_2t0);
6587 ((#4.0), x2, square_2t0);
6588 ((#4.0), x3, square_2t0);
6589 ((#4.0) , x4, (#4.0));
6590 (square_2t0, x5, square_2t0);
6591 ((square(#3.2)), x6, (square(#3.2)))
6594 ( (tau_0_x x1 x2 x3 x4 x5 x6 ) >. ((#0.205))))`;;
6596 (* interval verification by Ferguson *)
6599 Bound: 0.0890816152428
6601 Point: [3.99999999999, 3.99999999999, 3.99999999999, 3.99999999999, 10.2399999999, 3.99999999999]
6603 The inequality is OK in SPVI2002, but a sign error was introduced when it was
6604 copied to this file. The typo has been corrected.
6608 [((#4.0), x1, square_2t0);
6609 ((#4.0), x2, square_2t0);
6610 ((#4.0), x3, square_2t0);
6611 ((#4.0) , x4, (#4.0));
6612 ((#8.0) , x5, (square(#3.2)));
6613 ((#4.0) , x6, (#4.0) )
6616 ( (vor_0_x x1 x2 x3 x4 x5 x6 ) <. ((#0.009) +. ((sqrt x5)-(sqrt8))*(#0.14))))`;;
6618 let CKC_861511432= list_mk_conj[I_938003786;I_683897354;I_547486831;
6619 I_595674181;I_354217730; ];; (* kc group 18.9 *)
6622 LOC: 2002 k.c page 60
6627 (* interval verification by Ferguson *)
6628 (* CCC many false/infeasible. Cross diag constraint fixed 1/31/2008 *)
6629 let I_109046923_GEN=
6633 ((square ( # 3.2)) , x, (square_4t0))
6635 (((vor_0_x a1 a2 a4 x square_2t0 (#4.0) )+
6636 (vor_0_x a3 a2 a4 x (#4.0) (#8.0) )
6637 <. (--(#0.221))-(#0.0461))
6639 (cross_diag_x a1 a2 a4 x square_2t0 (#4.0) a3 (#4.0) (#8.0)
6641 (delta_x a1 a2 a4 x square_2t0 (#4.0) <. (#0.0)) \/
6642 (delta_x a3 a2 a4 x (#4.0) (#8.0) <. (#0.0)))))`;;
6644 (* interval verification by Ferguson *)
6647 (list_mk_comb(I_109046923_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
6649 (* interval verification by Ferguson *)
6652 (list_mk_comb(I_109046923_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
6654 (* interval verification by Ferguson *)
6657 (list_mk_comb(I_109046923_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
6659 (* interval verification by Ferguson *)
6662 (list_mk_comb(I_109046923_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
6664 (* interval verification by Ferguson *)
6667 (list_mk_comb(I_109046923_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
6669 (* interval verification by Ferguson *)
6672 Bound: 0.122198000542
6673 Point: [16.9619640963]
6674 My calculation of the cross-diag is < 3.2 which means that this
6675 isn't a counterexample.
6679 (list_mk_comb(I_109046923_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
6681 (* interval verification by Ferguson *)
6684 (list_mk_comb(I_109046923_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
6686 (* interval verification by Ferguson *)
6689 (list_mk_comb(I_109046923_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
6691 (* interval verification by Ferguson *)
6694 (list_mk_comb(I_109046923_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
6696 (* interval verification by Ferguson *)
6699 (list_mk_comb(I_109046923_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
6701 (* interval verification by Ferguson *)
6704 (list_mk_comb(I_109046923_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
6706 (* interval verification by Ferguson *)
6709 (list_mk_comb(I_109046923_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
6711 (* interval verification by Ferguson *)
6714 (list_mk_comb(I_109046923_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
6716 (* interval verification by Ferguson *)
6719 (list_mk_comb(I_109046923_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
6721 (* interval verification by Ferguson *)
6724 (list_mk_comb(I_109046923_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
6726 (* interval verification by Ferguson *)
6729 (list_mk_comb(I_109046923_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
6733 (* interval verification by Ferguson *)
6734 (* CCC many false/infeasible, cross diag fixed 1/31/2008 *)
6735 let I_642590101_GEN=
6739 ((square ( # 3.2)) , x, (square_4t0))
6741 (((tau_0_x a1 a2 a4 x square_2t0 (#4.0) )+
6742 (tau_0_x a3 a2 a4 x (#4.0) (#8.0) )
6745 (cross_diag_x a1 a2 a4 x square_2t0 (#4.0) a3 (#4.0) (#8.0)
6747 (delta_x a1 a2 a4 x square_2t0 (#4.0) <. (#0.0)) \/
6748 (delta_x a3 a2 a4 x (#4.0) (#8.0) <. (#0.0)))))`;;
6750 (* interval verification by Ferguson *)
6753 (list_mk_comb(I_642590101_GEN,[`#4.0`;`#4.0`;`#4.0`;`#4.0`]));;
6755 (* interval verification by Ferguson *)
6758 (list_mk_comb(I_642590101_GEN,[`#4.0`;`#4.0`;`#4.0`;`square_2t0`]));;
6760 (* interval verification by Ferguson *)
6763 (list_mk_comb(I_642590101_GEN,[`#4.0`;`#4.0`;`square_2t0`;`#4.0`]));;
6765 (* interval verification by Ferguson *)
6768 (list_mk_comb(I_642590101_GEN,[`#4.0`;`#4.0`;`square_2t0`;`square_2t0`]));;
6770 (* interval verification by Ferguson *)
6773 (list_mk_comb(I_642590101_GEN,[`#4.0`;`square_2t0`;`#4.0`;`#4.0`]));;
6775 (* interval verification by Ferguson *)
6777 Bound: 0.218305970844
6779 Point: [16.9397074241]
6780 My calculation of the cross-diag is < 3.2, so this is not a counterexample.
6784 (list_mk_comb(I_642590101_GEN,[`#4.0`;`square_2t0`;`#4.0`;`square_2t0`]));;
6786 (* interval verification by Ferguson *)
6789 (list_mk_comb(I_642590101_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`#4.0`]));;
6791 (* interval verification by Ferguson *)
6794 (list_mk_comb(I_642590101_GEN,[`#4.0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
6796 (* interval verification by Ferguson *)
6799 (list_mk_comb(I_642590101_GEN,[`square_2t0`;`#4.0`;`#4.0`;`#4.0`]));;
6801 (* interval verification by Ferguson *)
6804 (list_mk_comb(I_642590101_GEN,[`square_2t0`;`#4.0`;`#4.0`;`square_2t0`]));;
6806 (* interval verification by Ferguson *)
6809 (list_mk_comb(I_642590101_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`#4.0`]));;
6811 (* interval verification by Ferguson *)
6814 (list_mk_comb(I_642590101_GEN,[`square_2t0`;`#4.0`;`square_2t0`;`square_2t0`]));;
6816 (* interval verification by Ferguson *)
6819 (list_mk_comb(I_642590101_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`#4.0`]));;
6821 (* interval verification by Ferguson *)
6824 (list_mk_comb(I_642590101_GEN,[`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`]));;
6826 (* interval verification by Ferguson *)
6829 (list_mk_comb(I_642590101_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`]));;
6831 (* interval verification by Ferguson *)
6834 (list_mk_comb(I_642590101_GEN,[`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`]));;
6836 let CKC_746445726= list_mk_conj[
6837 I_642590101_16;I_642590101_15;I_642590101_14;I_642590101_13;I_642590101_12;
6838 I_642590101_11;I_642590101_10;I_642590101_9;I_642590101_8;I_642590101_7;
6839 I_642590101_6;I_642590101_5;I_642590101_4;I_642590101_3;I_642590101_2;
6840 I_642590101_1; I_109046923_16;I_109046923_15;I_109046923_14;I_109046923_13;
6841 I_109046923_12;I_109046923_11;I_109046923_10;I_109046923_9;I_109046923_8;
6842 I_109046923_7;I_109046923_6;I_109046923_5;I_109046923_4;I_109046923_3;
6843 I_109046923_2;I_109046923_1; ];; (* kc group 18.10 *)
6846 LOC: 2002 k.c page 60
6850 (* CCC Error: for much of this group a3 is not in scope here! Fixed 1/31/2008. *)
6851 (* interval verification by Ferguson *)
6852 let I_160800042_GEN=
6856 ((#8.0) , x, (square(#3.2)));
6857 ((#8.0) , x', (square(#3.2)))
6859 (((vor_0_x a2 (#4.0) a1 x (#4.0) (#4.0))+
6860 (vor_0_x a1 (#4.0) a5 x' square_2t0 x)+
6861 (vor_0_x a5 (#4.0) a4 (#4.0) (#4.0) x')
6864 (delta_x a2 (#4.0) a1 x (#4.0) (#4.0) <. (#0.0)) \/
6865 (delta_x a1 (#4.0) a5 x' square_2t0 x <. (#0.0)) \/
6866 (delta_x a5 (#4.0) a4 (#4.0) (#4.0) x' <. (#0.0)))))`;;
6868 (* interval verification by Ferguson *)
6871 (list_mk_comb(I_160800042_GEN,[`#4.0`;`#4.0`]));;
6873 (* interval verification by Ferguson *)
6876 (list_mk_comb(I_160800042_GEN,[`#4.0`;`square_2t0`]));;
6878 (* interval verification by Ferguson *)
6881 (list_mk_comb(I_160800042_GEN,[`square_2t0`;`#4.0`]));;
6883 (* interval verification by Ferguson *)
6886 (list_mk_comb(I_160800042_GEN,[`square_2t0`;`square_2t0`]));;
6888 (* interval verification by Ferguson *)
6889 let I_690272881_GEN=
6893 ((#8.0) , x, (square(#3.2)));
6894 ((#8.0) , x', (square(#3.2)))
6896 (((tau_0_x a2 (#4.0) a1 x (#4.0) (#4.0))+
6897 (tau_0_x a1 (#4.0) a5 x' square_2t0 x)+
6898 (tau_0_x a5 (#4.0) a4 (#4.0) (#4.0) x')
6901 (delta_x a2 (#4.0) a1 x (#4.0) (#4.0) <. (#0.0)) \/
6902 (delta_x a1 (#4.0) a5 x' square_2t0 x <. (#0.0)) \/
6903 (delta_x a5 (#4.0) a4 (#4.0) (#4.0) x' <. (#0.0)))))`;;
6905 (* interval verification by Ferguson *)
6908 (list_mk_comb(I_690272881_GEN,[`#4.0`;`#4.0`]));;
6910 (* interval verification by Ferguson *)
6913 (list_mk_comb(I_690272881_GEN,[`#4.0`;`square_2t0`]));;
6915 (* interval verification by Ferguson *)
6918 (list_mk_comb(I_690272881_GEN,[`square_2t0`;`#4.0`]));;
6920 (* interval verification by Ferguson *)
6923 (list_mk_comb(I_690272881_GEN,[`square_2t0`;`square_2t0`]));;
6925 let CKC_897046482= list_mk_conj[
6926 I_690272881_4;I_690272881_3;I_690272881_2;I_690272881_1
6927 ; I_160800042_4;I_160800042_3;I_160800042_2;I_160800042_1; ];; (* kc group 18.11 *)
6931 LOC: 2002 k.c page 60
6937 (* interval verification by Ferguson *)
6938 (* Note that this inequality only applies to a convex pentagon *)
6940 (* CCC many false/infeasible. I don't see any problem with it. Do you have a counterexample?
6941 In SPVI2002 there is a typo, but it appears to be correct in this file. *)
6942 let I_713930036_GEN=
6946 ((square(#3.2)),x,square_4t0);
6947 ((square(#3.2)),x',square_4t0)
6949 (((vor_0_x (#4.0) (#4.0) a1 x (#4.0) (#4.0))+
6950 (vor_0_x a1 (#4.0) a5 x' square_2t0 x)+
6951 (vor_0_x a5 (#4.0) (#4.0) (#4.0) (#4.0) x')
6954 ((dih_x (#4.0) a5 (#4.0) (#4.0) (#4.0) x') +
6955 (dih_x (#4.0) a5 a1 square_2t0 x x') +
6956 (dih_x (#4.0) (#4.0) a1 (#4.0) x (#4.0) ) < acs(--(&.53)/(&.75))) \/
6957 (delta_x (#4.0) (#4.0) a1 x (#4.0) (#4.0) <. (#0.0)) \/
6958 (delta_x a1 (#4.0) a5 x' square_2t0 x <. (#0.0)) \/
6959 (delta_x a5 (#4.0) (#4.0) (#4.0) (#4.0) x' <. (#0.0)))))`;;
6961 (* interval verification by Ferguson *)
6963 CCC false. See note on _4
6964 Bound: 0.0216447124442
6966 Point: [11.9999999941, 11.9999998104]
6971 (list_mk_comb(I_713930036_GEN,[`#4.0`;`#4.0`]));;
6973 (* interval verification by Ferguson *)
6975 CCC false. See note on _4
6976 Bound: 0.114998022539
6978 Point: [11.9999998616, 13.9200391298]
6983 (list_mk_comb(I_713930036_GEN,[`#4.0`;`square_2t0`]));;
6985 (* interval verification by Ferguson *)
6987 CCC false. See note on _4
6988 Bound: 0.114998022544
6990 Point: [13.9200391298, 11.9999998616]
6995 (list_mk_comb(I_713930036_GEN,[`square_2t0`;`#4.0`]));;
6997 (* interval verification by Ferguson *)
6999 Bound: 0.112874764386
7001 Point: [13.9200392672, 13.9200389776]
7003 Tom, I know you think this is not because of instability,
7004 but my calculations give
7005 [0.112294486983,1.91893398547,1.95123394064E~06,393.739050459,0.000453238439945]
7006 for the values of the respective functions.
7009 The arccos(-53/75) is approximately 2.35557.
7010 The left-hand side for that inequality is about 1.13184.
7011 (Two of the dihedrals are nearly zero because delta is about 0.
7012 The middle piece has dih 1.13184...)
7014 Here is my theory. I suspect you are still not switching between different
7015 formulas for dih on different parts of the domain, as you should be.
7016 This is causing your dihedral function to return an angle near pi,
7017 when it should be returning an angle near 0.
7019 Note that your constant 1.91893398547 + (2.3557 - 1.13184) is
7020 approximately 3.13562, which is suspiciously close to pi.
7022 Tom, It was worse than that. I didn't implement acos
7028 (list_mk_comb(I_713930036_GEN,[`square_2t0`;`square_2t0`]));;
7032 (* interval verification by Ferguson *)
7033 let I_724922588_GEN=
7037 ((square(#3.2)),x,square_4t0);
7038 ((square(#3.2)),x',square_4t0)
7040 (((tau_0_x (#4.0) (#4.0) a1 x (#4.0) (#4.0))+
7041 (tau_0_x a1 (#4.0) a5 x' square_2t0 x)+
7042 (tau_0_x a5 (#4.0) (#4.0) (#4.0) (#4.0) x')
7045 ((dih_x (#4.0) a5 (#4.0) (#4.0) (#4.0) x') +
7046 (dih_x (#4.0) a5 a1 square_2t0 x x') +
7047 (dih_x (#4.0) (#4.0) a1 (#4.0) x (#4.0) ) < acs(--(&.53)/(&.75))) \/
7048 (delta_x (#4.0) (#4.0) a1 x (#4.0) (#4.0) <. (#0.0)) \/
7049 (delta_x a1 (#4.0) a5 x' square_2t0 x <. (#0.0)) \/
7050 (delta_x a5 (#4.0) (#4.0) (#4.0) (#4.0) x' <. (#0.0)))))`;;
7052 (* interval verification by Ferguson *)
7055 (list_mk_comb(I_724922588_GEN,[`#4.0`;`#4.0`]));;
7057 (* interval verification by Ferguson *)
7060 (list_mk_comb(I_724922588_GEN,[`#4.0`;`square_2t0`]));;
7062 (* interval verification by Ferguson *)
7065 (list_mk_comb(I_724922588_GEN,[`square_2t0`;`#4.0`]));;
7067 (* interval verification by Ferguson *)
7070 (list_mk_comb(I_724922588_GEN,[`square_2t0`;`square_2t0`]));;
7072 let CKC_928952883= list_mk_conj[I_724922588_4;I_724922588_3;I_724922588_2;I_724922588_1;
7073 I_713930036_4;I_713930036_3;I_713930036_2;I_713930036_1; ];; (* kc group 18.12 *)
7076 LOC: 2002 k.c page 61
7080 (* interval verification by Ferguson *)
7081 (* cross_diag constraint fixed 1/31/2008,
7082 Fixed bug in third branch of vor_0_x and delta_x 2/1/2008.
7083 It is correctly stated in SPVI2002.
7086 let I_821730621_GEN=
7090 ((#8.0) , x, (square(#3.2)));
7091 (square(#3.2),x',square_4t0)
7093 (((vor_0_x (#4.0) a2 (#4.0) (#4.0) x (#4.0))+
7094 (vor_0_x (#4.0) (#4.0) a4 (#4.0) x' x)+
7095 (vor_0_x (#4.0) a4 a5 (#4.0) (square_2t0) x')
7098 (cross_diag_x (#4.0) (#4.0) a4 x' (#4.0) x a5 (#4.0) square_2t0
7100 (delta_x (#4.0) a2 (#4.0) (#4.0) x (#4.0) <. (#0.0)) \/
7101 (delta_x (#4.0) (#4.0) a4 (#4.0) x' x <. (#0.0)) \/
7102 (delta_x (#4.0) a4 a5 (#4.0) (square_2t0) x' <. (#0.0)))))`;;
7105 (* interval verification by Ferguson *)
7107 Bound: 0.189254861226
7109 Point: [10.0605373011, 11.9999998741]
7110 {y,yp} = {y, yp} = {10.0605373011, 11.9999998741} // Sqrt;
7111 CrossDiag[y1_, y2_, y3_, y4_, y5_, y6_, y7_, y8_, y9_] :=
7112 Enclosed[y1, y5, y6,
7113 y4, y2, y3, y7, y8, y9];
7114 CrossDiag[2, 2, 2, yp, 2, y, 2, 2, 2.51]; (* yields 3.28.. *)
7115 tt = {VorVc[2, 2, 2, 2, y, 2], VorVc[2, 2, 2, 2, yp, y], VorVc[2, 2, 2, 2, 2.51, yp]};
7118 {Delta[2, 2, 2, 2, y, 2], Delta[2, 2, 2, 2, yp, y], Delta[2, 2, 2, 2, 2.51, yp]};
7119 (* Yields {78.04, 143.98, 6.043*10^-6} *)
7125 (list_mk_comb(I_821730621_GEN,[`#4.0`;`#4.0`;`#4.0`]));;
7127 (* interval verification by Ferguson *)
7130 (list_mk_comb(I_821730621_GEN,[`#4.0`;`#4.0`;`square_2t0`]));;
7132 (* interval verification by Ferguson *)
7134 Bound: 0.0948377771411
7136 Point: [8.57185841044, 13.3519358538]
7140 (list_mk_comb(I_821730621_GEN,[`#4.0`;`square_2t0`;`#4.0`]));;
7142 (* interval verification by Ferguson *)
7145 (list_mk_comb(I_821730621_GEN,[`#4.0`;`square_2t0`;`square_2t0`]));;
7147 (* interval verification by Ferguson *)
7150 Bound: 0.177722784329
7152 Point: [9.69989999996, 11.9999999999]
7157 (list_mk_comb(I_821730621_GEN,[`square_2t0`;`#4.0`;`#4.0`]));;
7159 (* interval verification by Ferguson *)
7162 (list_mk_comb(I_821730621_GEN,[`square_2t0`;`#4.0`;`square_2t0`]));;
7164 (* interval verification by Ferguson *)
7167 (list_mk_comb(I_821730621_GEN,[`square_2t0`;`square_2t0`;`#4.0`]));;
7169 (* interval verification by Ferguson *)
7172 (list_mk_comb(I_821730621_GEN,[`square_2t0`;`square_2t0`;`square_2t0`]));;
7176 (* interval verification by Ferguson *)
7177 (* cross diag constraint fixed 1/31/2008 *)
7178 (* b5 edge length in tau_0_x and delta_x fixed 2/1/2008 *)
7180 let I_890642961_GEN=
7184 ((#8.0) , x, (square(#3.2)));
7185 (square(#3.2),x',square_4t0)
7187 (((tau_0_x (#4.0) a2 (#4.0) (#4.0) x (#4.0))+
7188 (tau_0_x (#4.0) (#4.0) a4 (#4.0) x' x)+
7189 (tau_0_x (#4.0) a4 a5 (#4.0) (square_2t0) x')
7192 (cross_diag_x (#4.0) (#4.0) a4 x' (#4.0) x a5 (#4.0) square_2t0
7194 (delta_x (#4.0) a2 (#4.0) (#4.0) x (#4.0) <. (#0.0)) \/
7195 (delta_x (#4.0) (#4.0) a4 (#4.0) x' x <. (#0.0)) \/
7196 (delta_x (#4.0) a4 a5 (#4.0) (square_2t0) x' <. (#0.0)))))`;;
7199 (* interval verification by Ferguson *)
7202 Bound: 0.282549826421
7204 Point: [9.27255301111, 11.9999999996];
7205 {y,yp} = {9.27255301111, 11.9999999996}//Sqrt;
7206 tt = {tauVc[2,2,2,2,y,2],tauVc[2,2,2,2,yp,y],tauVc[2,2,2,2,2.51,yp]}
7208 {Delta[2,2,2,2,y,2],Delta[2,2,2,2,yp,y],Delta[2,2,2,2,2.51,yp]}
7209 CrossDiagE[2,2,2,yp,2,y,2,2,2.51]
7213 (list_mk_comb(I_890642961_GEN,[`#4.0`;`#4.0`;`#4.0`]));;
7215 (* interval verification by Ferguson *)
7218 (list_mk_comb(I_890642961_GEN,[`#4.0`;`#4.0`;`square_2t0`]));;
7220 (* interval verification by Ferguson *)
7223 (list_mk_comb(I_890642961_GEN,[`#4.0`;`square_2t0`;`#4.0`]));;
7225 (* interval verification by Ferguson *)
7226 (* CCC See comments above
7229 Point: [9.26173984803, 11.7132329274]
7233 (list_mk_comb(I_890642961_GEN,[`#4.0`;`square_2t0`;`square_2t0`]));;
7235 (* interval verification by Ferguson *)
7236 (* CCC See comments above
7237 Bound: 0.245027755733
7239 Point: [9.42893490619, 11.9999999297]
7243 (list_mk_comb(I_890642961_GEN,[`square_2t0`;`#4.0`;`#4.0`]));;
7245 (* interval verification by Ferguson *)
7248 (list_mk_comb(I_890642961_GEN,[`square_2t0`;`#4.0`;`square_2t0`]));;
7250 (* interval verification by Ferguson *)
7251 (* CCC See comments above
7252 Bound: 0.00265356467075
7254 Point: [8.13556916171, 12.1086273347]
7258 (list_mk_comb(I_890642961_GEN,[`square_2t0`;`square_2t0`;`#4.0`]));;
7260 (* interval verification by Ferguson *)
7261 (* CCC See comments above
7262 Bound: 0.0405287948262
7264 Point: [9.69989999999, 11.7132329804]
7268 (list_mk_comb(I_890642961_GEN,[`square_2t0`;`square_2t0`;`square_2t0`]));;
7270 let CKC_673800906= list_mk_conj[I_890642961_8;I_890642961_7;I_890642961_6;I_890642961_5;
7271 I_890642961_4;I_890642961_3;I_890642961_2;I_890642961_1;
7272 I_821730621_8;I_821730621_7;I_821730621_6;I_821730621_5;
7273 I_821730621_4;I_821730621_3;I_821730621_2;I_821730621_1; ];; (* kc group 18.13 *)
7276 LOC: 2002 k.c page 60
7280 (* interval verification by Ferguson *)
7283 [((#4.0), x1, square_2t0);
7284 ((#4.0), x2, square_2t0);
7285 ((#4.0), x3, square_2t0);
7286 ((#8.0) , x4, (#8.0));
7287 (square_2t0 , x5, (#8.0) );
7288 (square_2t0 , x6, (#8.0) )
7290 (vor_0_x x1 x2 x3 x4 x5 x6 <. --(#0.168) - (#0.009))
7293 (* interval verification by Ferguson *)
7296 [((#4.0), x1, square_2t0);
7297 ((#4.0), x2, square_2t0);
7298 ((#4.0), x3, square_2t0);
7299 ((#8.0) , x4, (#8.0));
7300 (square_2t0 , x5, (#8.0) );
7301 (square_2t0 , x6, (#8.0) )
7303 (tau_0_x x1 x2 x3 x4 x5 x6 > (#0.352) - (#0.05925))
7306 let CKC_315678695= list_mk_conj[I_535906363;I_341667126; ];; (* kc group 18.14 *)
7309 LOC: 2002 k.c page 61
7314 CCC fail concerned about this one... Thanks for the concern, man.
7316 Bound: 0.0215663812919
7318 Point: [3.99999999999, 3.99999999999, 3.99999999999, 3.99999999999, 7.99999999999, 8]
7320 A typo in the constant fixed 1/31/2008.
7324 [((#4.0), x1, square_2t0);
7325 ((#4.0), x2, square_2t0);
7326 ((#4.0), x3, square_2t0);
7327 ((#4.0), x4, square_2t0);
7328 ((#8.0) , x5, square (#3.2));
7329 ((#8.0) , x6, square (#3.2))
7331 (vor_0_x x1 x2 x3 x4 x5 x6 <. --(#0.146) )
7337 [((#4.0), x1, square_2t0);
7338 ((#4.0), x2, square_2t0);
7339 ((#4.0), x3, square_2t0);
7340 ((#4.0), x4, square_2t0);
7341 ((#8.0) , x5, square (#3.2));
7342 ((#8.0) , x6, square (#3.2))
7344 (tau_0_x x1 x2 x3 x4 x5 x6 +
7345 (tau_0_x x1 (#4.0) x3 (#4.0) x5 (#4.0)) >. (#0.31) )
7351 [((#4.0), x1, square_2t0);
7352 ((#4.0), x2, square_2t0);
7353 ((#4.0), x3, square_2t0);
7354 ((#4.0), x4, square_2t0);
7355 ((#8.0) , x5, square (#3.2));
7356 ((#8.0) , x6, square (#3.2))
7358 (tau_0_x x1 x2 x3 x4 x5 x6 +
7359 (tau_0_x x1 square_2t0 x3 (#4.0) x5 (#4.0)) >. (#0.31) )
7362 let CKC_468742136= list_mk_conj[I_130008809_2;I_130008809_1;I_516537931; ];; (* kc group 18.15 *)
7365 LOC: 2002 k.c page 60
7371 [((#4.0), x1, square_2t0);
7372 ((#4.0), x2, square_2t0);
7373 ((#4.0), x3, square_2t0);
7374 ((#4.0), x4, square_2t0);
7375 (square_2t0 , x5, (#8.0) );
7376 ((#8.0) , x6, square (#3.2))
7378 (vor_0_x x1 x2 x3 x4 x5 x6 <. --(#0.084) )
7382 (* I_292827481 deprecated *)
7385 (* interval verification in partK.cc *)
7388 [((#4.0), x1, (#4.0) );
7389 ((#4.0), x2, square_2t0);
7390 ((#4.0), x3, (#4.0) );
7391 ((#4.0), x4, (#4.0) );
7392 ((#8.0) , x5, square (#3.2));
7393 ((#4.0), x6, (#4.0) )
7395 (vor_0_x x1 x2 x3 x4 x5 x6 < (#0.009) + ((sqrt x5 - sqrt8)*(#0.1)) )
7401 [((#4.0), x1, square_2t0);
7402 ((#4.0), x2, square_2t0);
7403 ((#4.0), x3, square_2t0);
7404 ((#4.0), x4, square_2t0);
7405 (square_2t0 , x5, (#8.0) );
7406 ((#8.0) , x6, square (#3.2))
7408 (tau_0_x x1 x2 x3 x4 x5 x6 >. (#0.176) )
7411 let CKC_938091791= list_mk_conj[I_286122364;I_710875528;(* I_292827481;*)I_531861442; ];; (* kc group 18.16 *)
7413 (* end of 2002:kc *)
7418 group hash codes spIV :
7424 Here are the composite inequalities
7425 for the various groups:
7475 LOC: 2002 IV page 46.
7480 It says we may assume y6=2, and equality is entered below in the bounds
7482 (* interval verification by Ferguson *)
7483 (* moved 757995764 to inequality_spec.ml *)
7500 (* interval verification by Ferguson *)
7501 (* moved 735258244 to inequality_spec.ml *)
7507 (* interval verification by Ferguson *)
7510 [((#4.0), x1, square_2t0);
7511 ((#4.0), x2, square_2t0);
7512 ((#4.0), x3, square_2t0);
7513 ((#4.0), x4, (square (#3.2)));
7515 (square_2t0, x5, square_2t0);
7516 (square_2t0, x6, square_2t0)
7519 (beta (arclength (sqrt x1) t0 (#1.6)) (arclength (sqrt x1) (sqrt x2) (sqrt x6))) <.
7520 (dih2_x x1 x2 x3 x4 x5 x6))`;;
7531 (* interval verification by Ferguson *)
7534 [((square (#2.2)), x1, square_2t0);
7535 ((#4.0), x2, square_2t0);
7536 ((#4.0), x3, square_2t0);
7537 ((#4.0), x4, (square (#3.2)));
7539 ((square (#3.2)), x5, (square (#3.2)));
7540 ((#4.0), x6, (#4.0))
7543 (beta (arclength (sqrt x1) t0 (#1.6)) (arclength (sqrt x1) (sqrt x2) (sqrt x6))) <.
7544 (dih2_x x1 x2 x3 x4 x5 x6))`;;
7548 (* interval verification by Ferguson *)
7552 ((#4.0), x1, (square (#2.2)));
7553 ((#4.0), x2, (square_2t0));
7554 (square_2t0, x3, square_2t0);
7555 ((square (#3.2)), x4, (square (#3.2)));
7556 ((square (#3.2)), x5, (square (#3.2)));
7557 ((#4.0), x6, (#4.0))
7559 (let y1 = (sqrt x1) in
7560 let y2 = (sqrt x2) in
7561 let psi = (arclength y1 t0 (#1.6)) in
7562 let eta126 = (eta_x x1 x2 x6) in
7563 ((dihR (y2/(&2)) eta126 (y1/(&.2 * cos(psi))))
7565 (dih2_x x1 x2 x3 x4 x5 x6)
7571 [((#4.0), x1, square_2t0);
7572 ((#4.0), x2, square_2t0);
7573 ((#4.0), x3, square_2t0);
7574 ((square (#2.77)), x4, (#8.0));
7576 ((#4.0), x5, square_2t0);
7577 ((#4.0), x6, square_2t0)
7580 ( (vort_x x1 x2 x3 x4 x5 x6 (#1.385)) <. (#0.00005)) \/
7581 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
7588 [((#4.0), x1, square_2t0);
7589 ((#4.0), x2, square_2t0);
7590 ((#4.0), x3, square_2t0);
7591 ((square (#2.77)), x4, (#8.0));
7593 ((#4.0), x5, square_2t0);
7594 ((#4.0), x6, square_2t0)
7597 ( (vort_x x1 x2 x3 x4 x5 x6 (#1.385)) <. (#0.00005)) \/
7598 ( (eta_x x2 x3 x4) <. (sqrt (#2.0))))`;;
7604 [((#4.0), x1, square_2t0);
7605 ((#4.0), x2, square_2t0);
7606 ((#4.0), x3, square_2t0);
7607 ((square (#2.77)), x4, (#8.0));
7609 ((#4.0), x5, square_2t0);
7610 ((#4.0), x6, square_2t0)
7613 ( (tauVt_x x1 x2 x3 x4 x5 x6 (#1.385)) >. (#0.0682)) \/
7614 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
7619 [((#4.0), x1, square_2t0);
7620 ((#4.0), x2, square_2t0);
7621 ((#4.0), x3, square_2t0);
7622 ((square (#2.77)), x4, (#8.0));
7624 ((#4.0), x5, square_2t0);
7625 ((#4.0), x6, square_2t0)
7628 ( (tauVt_x x1 x2 x3 x4 x5 x6 (#1.385)) >. (#0.0682)) \/
7629 ( (eta_x x2 x3 x4) <. (sqrt (#2.0))))`;;
7633 let CIVA1_193836552 = list_mk_conj [
7634 I_757995764;I_735258244;I_343330051;I_49446087;I_799187442 ;
7635 I_275706375;I_324536936;I_983547118;I_206278009;];;
7639 LOC: 2002 IV, page 46
7646 [(square_2t0, x1, (#8.0));
7647 ((#4.0), x2, square_2t0);
7648 ((#4.0), x3, square_2t0);
7649 ((#4.0), x4, square_2t0);
7651 ((#4.0), x5, square_2t0);
7652 ((#4.0), x6, square_2t0)
7654 ( (nu_x x1 x2 x3 x4 x5 x6) <. ( (--. (#4.3223)) +. ( (#4.10113) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
7660 [(square_2t0, x1, (#8.0));
7661 ((#4.0), x2, square_2t0);
7662 ((#4.0), x3, square_2t0);
7663 ((#4.0), x4, square_2t0);
7665 ((#4.0), x5, square_2t0);
7666 ((#4.0), x6, square_2t0)
7668 ( (nu_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.9871)) +. ( (#0.80449) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
7674 [(square_2t0, x1, (#8.0));
7675 ((#4.0), x2, square_2t0);
7676 ((#4.0), x3, square_2t0);
7677 ((#4.0), x4, square_2t0);
7679 ((#4.0), x5, square_2t0);
7680 ((#4.0), x6, square_2t0)
7682 ( (nu_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.8756)) +. ( (#0.70186) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
7688 [(square_2t0, x1, (#8.0));
7689 ((#4.0), x2, square_2t0);
7690 ((#4.0), x3, square_2t0);
7691 ((#4.0), x4, square_2t0);
7693 ((#4.0), x5, square_2t0);
7694 ((#4.0), x6, square_2t0)
7696 ( (nu_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.3404)) +. ( (#0.24573) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
7703 [(square_2t0, x1, (#8.0));
7704 ((#4.0), x2, square_2t0);
7705 ((#4.0), x3, square_2t0);
7706 ((#4.0), x4, square_2t0);
7708 ((#4.0), x5, square_2t0);
7709 ((#4.0), x6, square_2t0)
7711 ( (nu_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.0024)) +. ( (#0.00154) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
7717 [(square_2t0, x1, (#8.0));
7718 ((#4.0), x2, square_2t0);
7719 ((#4.0), x3, square_2t0);
7720 ((#4.0), x4, square_2t0);
7722 ((#4.0), x5, square_2t0);
7723 ((#4.0), x6, square_2t0)
7725 ( (nu_x x1 x2 x3 x4 x5 x6) <. ( (#0.1196) +. ( (--. (#0.07611)) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
7727 let CIVA2_815492935 = list_mk_conj [
7728 I_413688580;I_805296510;I_136610219;
7729 I_379204810;I_878731435;I_891740103;];;
7733 LOC: 2002 IV, page 46
7741 [(square_2t0, x1, (#8.0));
7742 ((#4.0), x2, square_2t0);
7743 ((#4.0), x3, square_2t0);
7744 ((#4.0), x4, square_2t0);
7746 ((#4.0), x5, square_2t0);
7747 ((#4.0), x6, square_2t0)
7749 ( (( --. ) (taunu_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#4.42873)) +. ( (#4.16523) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
7756 [(square_2t0, x1, (#8.0));
7757 ((#4.0), x2, square_2t0);
7758 ((#4.0), x3, square_2t0);
7759 ((#4.0), x4, square_2t0);
7761 ((#4.0), x5, square_2t0);
7762 ((#4.0), x6, square_2t0)
7764 ( (( --. ) (taunu_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#1.01104)) +. ( (#0.78701) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
7770 [(square_2t0, x1, (#8.0));
7771 ((#4.0), x2, square_2t0);
7772 ((#4.0), x3, square_2t0);
7773 ((#4.0), x4, square_2t0);
7775 ((#4.0), x5, square_2t0);
7776 ((#4.0), x6, square_2t0)
7778 ( (( --. ) (taunu_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#0.99937)) +. ( (#0.77627) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
7785 [(square_2t0, x1, (#8.0));
7786 ((#4.0), x2, square_2t0);
7787 ((#4.0), x3, square_2t0);
7788 ((#4.0), x4, square_2t0);
7790 ((#4.0), x5, square_2t0);
7791 ((#4.0), x6, square_2t0)
7793 ( (( --. ) (taunu_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#0.34877)) +. ( (#0.21916) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
7799 [(square_2t0, x1, (#8.0));
7800 ((#4.0), x2, square_2t0);
7801 ((#4.0), x3, square_2t0);
7802 ((#4.0), x4, square_2t0);
7804 ((#4.0), x5, square_2t0);
7805 ((#4.0), x6, square_2t0)
7807 ( (( --. ) (taunu_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#0.11434)) +. ( (#0.05107) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
7813 [(square_2t0, x1, (#8.0));
7814 ((#4.0), x2, square_2t0);
7815 ((#4.0), x3, square_2t0);
7816 ((#4.0), x4, square_2t0);
7818 ((#4.0), x5, square_2t0);
7819 ((#4.0), x6, square_2t0)
7821 ( (( --. ) (taunu_x x1 x2 x3 x4 x5 x6)) <. ( (#0.07749) +. ( (--. (#0.07106)) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
7823 let CIVA3_729988292 = list_mk_conj
7824 [ I_334002329;I_883139937;I_507989176;I_244435805;I_930176500;
7829 LOC: 2002 IV, page 47
7835 In this section and in section A5 we assumed dih_x ( <=. ) (#2.46)
7839 [(square_2t0, x1, (#8.0));
7840 ((#4.0), x2, square_2t0);
7841 ((#4.0), x3, square_2t0);
7842 (square_2t0, x4, (#8.0));
7844 ((#4.0), x5, square_2t0);
7845 ((#4.0), x6, square_2t0)
7848 ( (vorC0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#3.421)) +. ( (#2.28501) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
7849 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
7855 [(square_2t0, x1, (#8.0));
7856 ((#4.0), x2, square_2t0);
7857 ((#4.0), x3, square_2t0);
7858 (square_2t0, x4, (#8.0));
7860 ((#4.0), x5, square_2t0);
7861 ((#4.0), x6, square_2t0)
7864 ( (vorC0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#2.616)) +. ( (#1.67382) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
7865 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
7871 [(square_2t0, x1, (#8.0));
7872 ((#4.0), x2, square_2t0);
7873 ((#4.0), x3, square_2t0);
7874 (square_2t0, x4, (#8.0));
7876 ((#4.0), x5, square_2t0);
7877 ((#4.0), x6, square_2t0)
7880 ( (vorC0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#1.4486)) +. ( (#0.8285) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
7881 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
7887 [(square_2t0, x1, (#8.0));
7888 ((#4.0), x2, square_2t0);
7889 ((#4.0), x3, square_2t0);
7890 (square_2t0, x4, (#8.0));
7892 ((#4.0), x5, square_2t0);
7893 ((#4.0), x6, square_2t0)
7896 ( (vorC0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.79)) +. ( (#0.390925) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
7897 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
7903 [(square_2t0, x1, (#8.0));
7904 ((#4.0), x2, square_2t0);
7905 ((#4.0), x3, square_2t0);
7906 (square_2t0, x4, (#8.0));
7908 ((#4.0), x5, square_2t0);
7909 ((#4.0), x6, square_2t0)
7912 ( (vorC0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.3088)) +. ( (#0.12012) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
7913 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
7919 [(square_2t0, x1, (#8.0));
7920 ((#4.0), x2, square_2t0);
7921 ((#4.0), x3, square_2t0);
7922 (square_2t0, x4, (#8.0));
7924 ((#4.0), x5, square_2t0);
7925 ((#4.0), x6, square_2t0)
7928 ( (vorC0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.1558)) +. ( (#0.0501) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
7929 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
7931 let CIVA4_531888597= list_mk_conj
7932 [ I_649592321;I_600996944;I_70667639;I_99182343;I_578762805;
7936 LOC: 2002 IV, page 47
7942 ?comment at the beginning of the section
7944 not indicated in file
7949 [(square_2t0, x1, (#8.0));
7950 ((#4.0), x2, square_2t0);
7951 ((#4.0), x3, square_2t0);
7952 (square_2t0, x4, (#8.0));
7954 ((#4.0), x5, square_2t0);
7955 ((#4.0), x6, square_2t0)
7958 ( (( --. ) (tauC0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#3.3407)) +. ( (#2.1747) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
7959 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
7965 [(square_2t0, x1, (#8.0));
7966 ((#4.0), x2, square_2t0);
7967 ((#4.0), x3, square_2t0);
7968 (square_2t0, x4, (#8.0));
7970 ((#4.0), x5, square_2t0);
7971 ((#4.0), x6, square_2t0)
7974 ( (( --. ) (tauC0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#2.945)) +. ( (#1.87427) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
7975 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
7981 [(square_2t0, x1, (#8.0));
7982 ((#4.0), x2, square_2t0);
7983 ((#4.0), x3, square_2t0);
7984 (square_2t0, x4, (#8.0));
7986 ((#4.0), x5, square_2t0);
7987 ((#4.0), x6, square_2t0)
7990 ( (( --. ) (tauC0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#1.5035)) +. ( (#0.83046) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
7991 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
7997 [(square_2t0, x1, (#8.0));
7998 ((#4.0), x2, square_2t0);
7999 ((#4.0), x3, square_2t0);
8000 (square_2t0, x4, (#8.0));
8002 ((#4.0), x5, square_2t0);
8003 ((#4.0), x6, square_2t0)
8006 ( (( --. ) (tauC0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#1.0009)) +. ( (#0.48263) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8007 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8014 [(square_2t0, x1, (#8.0));
8015 ((#4.0), x2, square_2t0);
8016 ((#4.0), x3, square_2t0);
8017 (square_2t0, x4, (#8.0));
8019 ((#4.0), x5, square_2t0);
8020 ((#4.0), x6, square_2t0)
8023 ( (( --. ) (tauC0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#0.7787)) +. ( (#0.34833) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8024 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8030 [(square_2t0, x1, (#8.0));
8031 ((#4.0), x2, square_2t0);
8032 ((#4.0), x3, square_2t0);
8033 (square_2t0, x4, (#8.0));
8035 ((#4.0), x5, square_2t0);
8036 ((#4.0), x6, square_2t0)
8039 ( (( --. ) (tauC0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#0.4475)) +. ( (#0.1694) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8040 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8046 [(square_2t0, x1, (#8.0));
8047 ((#4.0), x2, square_2t0);
8048 ((#4.0), x3, square_2t0);
8049 (square_2t0, x4, (#8.0));
8051 ((#4.0), x5, square_2t0);
8052 ((#4.0), x6, square_2t0)
8055 ( (( --. ) (tauC0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#0.2568)) +. ( (#0.0822) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8056 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8059 let CIVA5_628964355= list_mk_conj
8060 [ I_719735900;I_359616783;I_440833181;I_578578364;I_327398152;
8061 I_314861952;I_234753056;];;
8064 LOC: 2002 IV, page 47
8069 In this section and in section A7 we assumed dih_x ( <=. ) (#2.46)
8075 [(square_2t0, x1, (#8.0));
8076 ((#4.0), x2, square_2t0);
8077 ((#4.0), x3, square_2t0);
8078 ((#8.0), x4, (square (#3.2)));
8080 ((#4.0), x5, square_2t0);
8081 ((#4.0), x6, square_2t0)
8084 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#3.58)) +. ( (#2.28501) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8085 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8091 [(square_2t0, x1, (#8.0));
8092 ((#4.0), x2, square_2t0);
8093 ((#4.0), x3, square_2t0);
8094 ((#8.0), x4, (square (#3.2)));
8096 ((#4.0), x5, square_2t0);
8097 ((#4.0), x6, square_2t0)
8100 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#2.715)) +. ( (#1.67382) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8101 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8107 [(square_2t0, x1, (#8.0));
8108 ((#4.0), x2, square_2t0);
8109 ((#4.0), x3, square_2t0);
8110 ((#8.0), x4, (square (#3.2)));
8112 ((#4.0), x5, square_2t0);
8113 ((#4.0), x6, square_2t0)
8116 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#1.517)) +. ( (#0.8285) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8117 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8123 [(square_2t0, x1, (#8.0));
8124 ((#4.0), x2, square_2t0);
8125 ((#4.0), x3, square_2t0);
8126 ((#8.0), x4, (square (#3.2)));
8128 ((#4.0), x5, square_2t0);
8129 ((#4.0), x6, square_2t0)
8132 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.858)) +. ( (#0.390925) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8133 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8139 [(square_2t0, x1, (#8.0));
8140 ((#4.0), x2, square_2t0);
8141 ((#4.0), x3, square_2t0);
8142 ((#8.0), x4, (square (#3.2)));
8144 ((#4.0), x5, square_2t0);
8145 ((#4.0), x6, square_2t0)
8148 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.358)) +. ( (#0.12012) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8149 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8155 [(square_2t0, x1, (#8.0));
8156 ((#4.0), x2, square_2t0);
8157 ((#4.0), x3, square_2t0);
8158 ((#8.0), x4, (square (#3.2)));
8160 ((#4.0), x5, square_2t0);
8161 ((#4.0), x6, square_2t0)
8164 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.186)) +. ( (#0.0501) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8165 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8168 let CIVA6_934150983 = list_mk_conj
8169 [ I_555481748;I_615152889;I_647971645;I_516606403;I_690552204;
8174 LOC: 2002 IV, page 47
8181 [(square_2t0, x1, (#8.0));
8182 ((#4.0), x2, square_2t0);
8183 ((#4.0), x3, square_2t0);
8184 ((#8.0), x4, (square (#3.2)));
8186 ((#4.0), x5, square_2t0);
8187 ((#4.0), x6, square_2t0)
8190 ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#3.48)) +. ( (#2.1747) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8191 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8197 [(square_2t0, x1, (#8.0));
8198 ((#4.0), x2, square_2t0);
8199 ((#4.0), x3, square_2t0);
8200 ((#8.0), x4, (square (#3.2)));
8202 ((#4.0), x5, square_2t0);
8203 ((#4.0), x6, square_2t0)
8206 ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#3.06)) +. ( (#1.87427) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8207 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8213 [(square_2t0, x1, (#8.0));
8214 ((#4.0), x2, square_2t0);
8215 ((#4.0), x3, square_2t0);
8216 ((#8.0), x4, (square (#3.2)));
8218 ((#4.0), x5, square_2t0);
8219 ((#4.0), x6, square_2t0)
8222 ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#1.58)) +. ( (#0.83046) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8223 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8229 [(square_2t0, x1, (#8.0));
8230 ((#4.0), x2, square_2t0);
8231 ((#4.0), x3, square_2t0);
8232 ((#8.0), x4, (square (#3.2)));
8234 ((#4.0), x5, square_2t0);
8235 ((#4.0), x6, square_2t0)
8238 ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#1.06)) +. ( (#0.48263) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8239 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8245 [(square_2t0, x1, (#8.0));
8246 ((#4.0), x2, square_2t0);
8247 ((#4.0), x3, square_2t0);
8248 ((#8.0), x4, (square (#3.2)));
8250 ((#4.0), x5, square_2t0);
8251 ((#4.0), x6, square_2t0)
8254 ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#0.83)) +. ( (#0.34833) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8255 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8261 [(square_2t0, x1, (#8.0));
8262 ((#4.0), x2, square_2t0);
8263 ((#4.0), x3, square_2t0);
8264 ((#8.0), x4, (square (#3.2)));
8266 ((#4.0), x5, square_2t0);
8267 ((#4.0), x6, square_2t0)
8270 ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#0.50)) +. ( (#0.1694) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8271 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8277 [(square_2t0, x1, (#8.0));
8278 ((#4.0), x2, square_2t0);
8279 ((#4.0), x3, square_2t0);
8280 ((#8.0), x4, (square (#3.2)));
8282 ((#4.0), x5, square_2t0);
8283 ((#4.0), x6, square_2t0)
8286 ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) <. ( (--. (#0.29)) +. ( (#0.0822) *. (dih_x x1 x2 x3 x4 x5 x6)))) \/
8287 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.46)))`;;
8290 let CIVA7_187932932= list_mk_conj
8291 [ I_679673664;I_926514235;I_459744700;I_79400832;I_277388353;
8292 I_839852751;I_787458652;];;
8296 LOC: 2002 IV, page 47
8301 Need upper bound for y4 in all equations in this section
8302 Change so that each y4 is equality.
8308 [(square_2t0, x1, (#8.0));
8309 ((#4.0), x2, square_2t0);
8310 ((#4.0), x3, square_2t0);
8311 (square_2t0, x4, square_2t0);
8313 ((#4.0), x5, square_2t0);
8314 ((#4.0), x6, square_2t0)
8316 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.23))`;;
8322 [(square_2t0, x1, (#8.0));
8323 ((#4.0), x2, square_2t0);
8324 ((#4.0), x3, square_2t0);
8325 ((#8.0), x4, (#8.0));
8327 ((#4.0), x5, square_2t0);
8328 ((#4.0), x6, square_2t0)
8330 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.4167))`;;
8336 [(square_2t0, x1, (#8.0));
8337 ((#4.0), x2, square_2t0);
8338 ((#4.0), x3, square_2t0);
8339 ((square (#3.2)), x4, (square (#3.2)));
8341 ((#4.0), x5, square_2t0);
8342 ((#4.0), x6, square_2t0)
8344 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.65))`;;
8350 [(square_2t0, x1, (#8.0));
8351 ((#4.0), x2, square_2t0);
8352 ((#4.0), x3, square_2t0);
8353 ((#4.0), x4, (#4.0));
8355 ((#4.0), x5, square_2t0);
8356 ((#4.0), x6, square_2t0)
8358 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#0.956))`;;
8364 [(square_2t0, x1, (#8.0));
8365 ((#4.0), x2, square_2t0);
8366 ((#4.0), x3, square_2t0);
8367 ((#4.0), x4, (#4.0));
8369 ((#4.0), x5, (#8.0));
8370 ((#4.0), x6, square_2t0)
8372 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#0.28))`;;
8378 [((square (#2.7)), x1, (#8.0));
8379 ((#4.0), x2, square_2t0);
8380 ((#4.0), x3, square_2t0);
8381 ((square (#3.2)), x4, (square (#3.2)));
8383 ((#4.0), x5, square_2t0);
8384 ((#4.0), x6, square_2t0)
8386 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.714))`;;
8392 [(square_2t0, x1, (square (#2.7)));
8393 ((#4.0), x2, (square (#2.25)));
8394 ((#4.0), x3, square_2t0);
8396 ((square (#3.2)), x4, (square (#3.2)));
8397 ((#4.0), x5, square_2t0);
8398 ((#4.0), x6, square_2t0)
8400 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.714))`;;
8406 [(square_2t0, x1, (#8.0));
8407 ((#4.0), x2, square_2t0);
8408 ((#4.0), x3, square_2t0);
8409 ((#4.0), x4, square_2t0);
8411 ((#4.0), x5, square_2t0);
8412 ((#4.0), x6, square_2t0)
8414 ( (dih_x x1 x2 x3 x4 x5 x6) <. (#2.184))`;;
8416 let CIVA8_83777706= list_mk_conj
8417 [ I_499014780;I_901845849;I_410091263;I_125103581;I_504968542;
8418 I_770716154;I_666090270;I_971555266;];;
8421 LOC: 2002 IV, page 47--48
8426 (* interval verification by Ferguson *)
8427 (* Uses monotonoicity in x4 variable *)
8430 [((square (#2.696)), x1, (#8.0));
8431 ((square (#2.45)), x2, square_2t0);
8432 ((#4.0), x3, square_2t0);
8434 ((square (#2.77)), x4, (square (#2.77)));
8435 ((#4.0), x5, square_2t0);
8436 ((square (#2.45)), x6, square_2t0)
8438 ( (kappa (sqrt x1) (sqrt x2) (sqrt x3) (sqrt x4) (sqrt x5) (sqrt x6)) <. (--. (#0.003521)))`;;
8442 (* interval verification by Ferguson *)
8445 [(square_2t0, x1, (square (#2.696)));
8446 ((#4.0), x2, square_2t0);
8447 ((#4.0), x3, square_2t0);
8448 ((square (#2.77)), x4, (#8.0));
8449 ((#4.0), x5, square_2t0);
8450 ((#4.0), x6, square_2t0)
8453 ( (kappa (sqrt x1) (sqrt x2) (sqrt x3) (sqrt x4) (sqrt x5) (sqrt x6)) <. (--. (#0.017))) \/
8454 ( (eta_x x2 x3 x4) <. (sqrt (#2.0))))`;;
8458 (* interval verification by Ferguson *)
8461 [(square_2t0, x1, (square (#2.696)));
8462 ((#4.0), x2, square_2t0);
8463 ((#4.0), x3, square_2t0);
8465 ((square (#2.77)), x4, (#8.0));
8466 ((#4.0), x5, square_2t0);
8467 ((#4.0), x6, square_2t0)
8470 ( (kappa (sqrt x1) (sqrt x2) (sqrt x3) (sqrt x4) (sqrt x5) (sqrt x6)) <. (--. (#0.017))) \/
8471 ( (eta_x x4 x5 x6) <. (sqrt (#2.0))))`;;
8476 Equality has been assumed with x4 term
8478 (* interval verification by Ferguson *)
8481 [((square (#2.57)), x1, (#8.0));
8482 ((#4.0), x2, square_2t0);
8483 ((#4.0), x3, square_2t0);
8484 ((square (#3.2)), x4, (square (#3.2)));
8485 ((#4.0), x5, square_2t0);
8486 ((#4.0), x6, square_2t0)
8489 ( (kappa (sqrt x1) (sqrt x2) (sqrt x3) (sqrt x4) (sqrt x5) (sqrt x6)) <. (--. (#0.02274))) \/
8490 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
8495 Equality has been assumed with x4 term
8497 (* interval verification by Ferguson *)
8500 [(square_2t0, x1, (square (#2.57)));
8501 ((#4.0), x2, square_2t0);
8502 ((#4.0), x3, square_2t0);
8503 ((square (#3.2)), x4, (square (#3.2)));
8504 ((#4.0), x5, square_2t0);
8505 ((#4.0), x6, square_2t0)
8508 ( (kappa (sqrt x1) (sqrt x2) (sqrt x3) (sqrt x4) (sqrt x5) (sqrt x6)) <. (--. (#0.029))) \/
8509 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
8514 Equality has been assumed with x4 term
8516 (* interval verification by Ferguson *)
8519 [(square_2t0, x1, (square (#2.57)));
8520 ((#4.0), x2, (square (#2.25)));
8521 ((#4.0), x3, (square (#2.25)));
8523 ((square (#3.2)), x4, (square (#3.2)));
8524 ((#4.0), x5, (square (#2.25)));
8525 ((#4.0), x6, (square (#2.25)))
8528 ( (kappa (sqrt x1) (sqrt x2) (sqrt x3) (sqrt x4) (sqrt x5) (sqrt x6)) <. (--. (#0.03883))) \/
8529 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
8534 (* interval verification by Ferguson *)
8537 [(square_2t0, x1, (square (#2.57)));
8538 ((#4.0), x2, (square (#2.25)));
8539 ((#4.0), x3, (square (#2.25)));
8541 ((square (#3.2)), x4, (square (#3.2)));
8542 ((#4.0), x5, (square (#2.25)));
8543 ((#4.0), x6, square_2t0)
8546 ( (kappa (sqrt x1) (sqrt x2) (sqrt x3) (sqrt x4) (sqrt x5) (sqrt x6)) <. (--. (#0.0325))) \/
8547 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
8550 let CIVA9_618205535= list_mk_conj
8551 [ I_956875054;I_664200787;I_390273147;I_654422246;I_366536370;
8552 I_62532125;I_370631902;];;
8556 LOC: 2002 IV, page 48
8563 [((square (#2.696)), x1, (#8.0));
8564 ((#4.0), x2, square_2t0);
8565 ((#4.0), x3, square_2t0);
8566 ((#4.0), x4, square_2t0);
8568 ((#4.0), x5, square_2t0);
8569 ((#4.0), x6, square_2t0)
8571 ( (gamma_x x1 x2 x3 x4 x5 x6) <. (octavor0_x x1 x2 x3 x4 x5 x6))`;;
8578 [(square_2t0, x1, (#8.0));
8579 ((#4.0), x2, square_2t0);
8580 ((#4.0), x3, square_2t0);
8581 ((#4.0), x4, square_2t0);
8583 ((#4.0), x5, square_2t0);
8584 ((#4.0), x6, square_2t0)
8586 ( (gamma_x x1 x2 x3 x4 x5 x6) <. ( (octavor0_x x1 x2 x3 x4 x5 x6) +. (#0.01561)))`;;
8593 [((square (#2.57)), x1, (#8.0));
8594 ((#4.0), x2, square_2t0);
8595 ((#4.0), x3, square_2t0);
8596 ((#4.0), x4, square_2t0);
8598 ((#4.0), x5, square_2t0);
8599 ((#4.0), x6, square_2t0)
8601 ( (gamma_x x1 x2 x3 x4 x5 x6) <. ( (octavor0_x x1 x2 x3 x4 x5 x6) +. (#0.00935)))`;;
8608 [(square_2t0, x1, (square (#2.57)));
8609 ((square (#2.25)), x2, square_2t0);
8610 ((#4.0), x3, square_2t0);
8612 ((#4.0), x4, square_2t0);
8613 ((#4.0), x5, square_2t0);
8614 ((#4.0), x6, square_2t0)
8616 ( (gamma_x x1 x2 x3 x4 x5 x6) <. ( (octavor0_x x1 x2 x3 x4 x5 x6) +. (#0.00928)))`;;
8622 [(square_2t0, x1, (square (#2.57)));
8623 ((square (#2.25)), x2, square_2t0);
8624 ((#4.0), x3, square_2t0);
8626 ((#4.0), x4, square_2t0);
8627 ((#4.0), x5, square_2t0);
8628 ((square (#2.25)), x6, square_2t0)
8630 ( (gamma_x x1 x2 x3 x4 x5 x6) <. (octavor0_x x1 x2 x3 x4 x5 x6))`;;
8632 let CIVA10_73974037= list_mk_conj
8633 [ I_214637273;I_751772680;I_366146051;I_675766140;I_520734758;];;
8636 LOC: 2002 IV, page 48
8643 [((square (#2.696)), x1, (#8.0));
8644 ((#4.0), x2, (square (#2.45)));
8645 ((#4.0), x3, (square (#2.45)));
8647 ((#4.0), x4, square_2t0);
8648 ((#4.0), x5, square_2t0);
8649 ((#4.0), x6, square_2t0)
8651 ( (octavor_analytic_x x1 x2 x3 x4 x5 x6) <. (octavor0_x x1 x2 x3 x4 x5 x6))`;;
8658 [((square (#2.696)), x1, (#8.0));
8659 ((square (#2.45)), x2, square_2t0);
8660 ((#4.0), x3, square_2t0);
8662 ((#4.0), x4, square_2t0);
8663 ((square (#2.45)), x5, square_2t0);
8664 ((#4.0), x6, square_2t0)
8666 ( (octavor_analytic_x x1 x2 x3 x4 x5 x6) <. (octavor0_x x1 x2 x3 x4 x5 x6))`;;
8673 [(square_2t0, x1, (#8.0));
8674 ((#4.0), x2, square_2t0);
8675 ((#4.0), x3, square_2t0);
8676 ((#4.0), x4, square_2t0);
8678 ((#4.0), x5, square_2t0);
8679 ((#4.0), x6, square_2t0)
8681 ( (vor_analytic_x x1 x2 x3 x4 x5 x6) <. ( (vor_0_x x1 x2 x3 x4 x5 x6) +. (#0.003521)))`;;
8688 [((square (#2.696)), x1, (#8.0));
8689 ((square (#2.45)), x2, square_2t0);
8690 ((#4.0), x3, square_2t0);
8692 (square_2t0, x4, (square (#2.77)));
8693 ((#4.0), x5, square_2t0);
8694 ((square (#2.45)), x6, square_2t0)
8696 ( (vor_analytic_x x1 x2 x3 x4 x5 x6) <. ( (vor_0_x x1 x2 x3 x4 x5 x6) +. (--. (#0.003521))))`;;
8703 [(square_2t0, x1, (square (#2.696)));
8704 ((#4.0), x2, square_2t0);
8705 ((#4.0), x3, square_2t0);
8706 (square_2t0, x4, (#8.0));
8708 ((#4.0), x5, square_2t0);
8709 ((#4.0), x6, square_2t0)
8711 ( (vor_analytic_x x1 x2 x3 x4 x5 x6) <. ( (vor_0_x x1 x2 x3 x4 x5 x6) +. (--. (#0.009))))`;;
8718 [(square_2t0, x1, (square (#2.57)));
8719 ((#4.0), x2, square_2t0);
8720 ((#4.0), x3, square_2t0);
8721 ((#4.0), x4, square_2t0);
8723 ((#4.0), x5, square_2t0);
8724 ((#4.0), x6, square_2t0)
8727 ( (octavor_analytic_x x1 x2 x3 x4 x5 x6) <. (octavor0_x x1 x2 x3 x4 x5 x6)) \/
8728 ( (eta_x x1 x2 x6) <. (sqrt (#2.0))))`;;
8735 [(square_2t0, x1, (#8.0));
8736 ((#4.0), x2, square_2t0);
8737 ((#4.0), x3, (square (#2.2)));
8738 ((#4.0), x4, square_2t0);
8740 ((#4.0), x5, square_2t0);
8741 ((#4.0), x6, square_2t0)
8744 ( (octavor_analytic_x x1 x2 x3 x4 x5 x6) <. ( (octavor0_x x1 x2 x3 x4 x5 x6) +. (--. (#0.004131)))) \/
8745 ( (eta_x x1 x2 x6) >. (sqrt (#2.0))) \/
8746 ( (eta_x x1 x3 x5) <. (sqrt (#2.0))))`;;
8748 let CIVA11_764978100= list_mk_conj
8749 [ I_378432183;I_572206659;I_310679005;I_284970880;I_972111620;
8750 I_875762896;I_385332676;];;
8754 LOC: 2002 IV, page 48
8759 (* interval verification by Ferguson *)
8762 [(square_2t0, x1, (#8.0));
8763 (square_2t0, x2, (#8.0));
8764 ((#4.0), x3, square_2t0);
8765 ((#4.0), x4, square_2t0);
8767 ((#4.0), x5, square_2t0);
8768 ((#4.0), x6, square_2t0)
8771 ( (tau_analytic_x x1 x2 x3 x4 x5 x6) >.
8772 ( (#0.13) +. ( (#0.2) *. ( (dih_x x1 x2 x3 x4 x5 x6) +. ( pi / (--. (#2.0))))))) \/
8773 ( (eta_x x1 x2 x6) >. (sqrt (#2.0))))`;;
8778 (* interval verification by Ferguson *)
8781 [(square_2t0, x1, (#8.0));
8782 (square_2t0, x2, (#8.0));
8783 ((#4.0), x3, square_2t0);
8784 ((#4.0), x4, square_2t0);
8786 ((#4.0), x5, square_2t0);
8787 ((#4.0), x6, square_2t0)
8790 ( (tauVt_x x1 x2 x3 x4 x5 x6 (sqrt (#2.0))) >.
8791 ( (#0.13) +. ( (#0.2) *. ( (dih_x x1 x2 x3 x4 x5 x6) +. ( pi / (--. (#2.0))))))) \/
8792 ( (eta_x x1 x2 x6) <. (sqrt (#2.0))))`;;
8799 [((square (#2.75)), x1, (#8.0));
8800 ((#4.0), x2, square_2t0);
8801 ((#4.0), x3, square_2t0);
8802 ((#4.0), x4, square_2t0);
8804 ((#4.0), x5, square_2t0);
8805 ((#4.0), x6, square_2t0)
8807 ( (nu_x x1 x2 x3 x4 x5 x6) <. ( (--. (#0.3429)) +. ( (#0.24573) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
8813 Equality used in dih_x equation
8817 [(square_2t0, x1, (square (#2.75)));
8818 ((#4.0), x2, square_2t0);
8819 ((#4.0), x3, square_2t0);
8820 (square_2t0, x4, (#8.0));
8822 ((#4.0), x5, square_2t0);
8823 ((#4.0), x6, square_2t0)
8826 ( (vorC0_x x1 x2 x3 x4 x5 x6) <. (--. (#0.0571))) \/
8827 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#2.2)))`;;
8829 let CIVA12_855294746= list_mk_conj
8830 [ I_970291025;I_524345535;I_812894433;I_404793781;];;
8834 LOC: 2002 IV, page 48--49
8841 [(square_2t0, x1, (#8.0));
8842 ((#4.0), x2, square_2t0);
8843 ((#4.0), x3, square_2t0);
8844 ((#4.0), x4, square_2t0);
8846 ((#4.0), x5, square_2t0);
8847 ((#4.0), x6, square_2t0)
8849 ( (taunu_x x1 x2 x3 x4 x5 x6) >. (#0.033))`;;
8856 [((#4.0), x1, square_2t0);
8857 ((#4.0), x2, square_2t0);
8858 ((#4.0), x3, square_2t0);
8859 ((#8.0), x4, (#8.0));
8861 ((#4.0), x5, square_2t0);
8862 ((#4.0), x6, square_2t0)
8864 ( (tau_0_x x1 x2 x3 x4 x5 x6) >. ( (#0.06585) +. (--. (#0.0066))))`;;
8871 [((#4.0), x1, square_2t0);
8872 ((#4.0), x2, square_2t0);
8873 ((#4.0), x3, square_2t0);
8874 ((#8.0), x4, (#8.0));
8876 ((#4.0), x5, square_2t0);
8877 ((#4.0), x6, square_2t0)
8879 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. (#0.009))`;;
8885 [((#4.0), x2, square_2t0);
8886 ((#4.0), x3, square_2t0);
8887 ((#8.0), x4, (square (#3.2)))
8889 ( (vor_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0)) <. (#0.0461))`;;
8894 Weak inequality used ( <=. ) in next one below
8898 [((#4.0), x3, square_2t0);
8899 ((#8.0), x4, (square (#3.2)))
8901 ( (vor_0_x square_2t0 (#4.0) x3 x4 (#4.0) (#4.0)) <=. (#0.0))`;;
8907 [((#4.0), x1, square_2t0);
8908 ((#4.0), x2, square_2t0);
8909 ((#8.0), x4, (square (#3.2)))
8911 ( (vor_0_x x1 x2 square_2t0 x4 (#4.0) (#4.0)) <. (#0.0))`;;
8917 [((#4.0), x2, square_2t0);
8918 ((#4.0), x3, square_2t0);
8919 ((#8.0), x4, (square (#3.2)))
8921 ( (tau_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0)) >. (#0.014))`;;
8926 Inequality used ( >=. ) in next one
8930 [((#8.0), x4, (square (#3.2)))
8932 ( (tau_0_x square_2t0 (#4.0) (#4.0) x4 (#4.0) (#4.0)) >=. (#0.0))`;;
8938 [((#4.0), x1, square_2t0);
8939 ((#4.0), x2, square_2t0);
8940 ((#8.0), x4, (square (#3.2)))
8942 ( (tau_0_x x1 x2 square_2t0 x4 (#4.0) (#4.0)) >. (#0.06585))`;;
8947 (* interval verification in partK.cc *)
8950 [((square (#2.696)), x1, (#8.0));
8951 ((#4.0), x2, square_2t0);
8952 ((#4.0), x3, square_2t0);
8953 ((#4.0), x4, square_2t0);
8955 ((#4.0), x5, square_2t0);
8956 ((#4.0), x6, square_2t0)
8958 ( (nu_x x1 x2 x3 x4 x5 x6) <.
8959 ( (vor_0_x x1 x2 x3 x4 x5 x6) +. ( (#0.01) *. ( ( pi / (#2.0)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6))))))`;;
8963 (* interval verification in partK.cc *)
8966 [((square (#2.6)), x1, (square (#2.696)));
8967 ((#4.0), x2, square_2t0);
8968 ((#4.0), x3, square_2t0);
8970 ((square (#2.1)), x4, square_2t0);
8971 ((#4.0), x5, square_2t0);
8972 ((#4.0), x6, square_2t0)
8974 ( (nu_x x1 x2 x3 x4 x5 x6) <. (vor_0_x x1 x2 x3 x4 x5 x6))`;;
8978 (* interval verification in partK.cc *)
8981 [((#4.0), x1, square_2t0);
8982 ((#4.0), x2, square_2t0);
8983 ((#4.0), x3, square_2t0);
8985 (square_2t0, x4, (#8.0));
8986 ((#4.0), x5, square_2t0);
8987 ((#4.0), x6, square_2t0)
8989 ( (mu_flat_x x1 x2 x3 x4 x5 x6) <. ( (vor_0_x x1 x2 x3 x4 x5 x6) +. (#0.0268)))`;;
8994 (* interval verification in partK.cc *)
8997 [((#4.0), x1, (square (#2.17)));
8998 ((#4.0), x2, square_2t0);
8999 ((#4.0), x3, square_2t0);
9001 (square_2t0, x4, (#8.0));
9002 ((#4.0), x5, square_2t0);
9003 ((#4.0), x6, square_2t0)
9005 ( (mu_flat_x x1 x2 x3 x4 x5 x6) <. ( (vor_0_x x1 x2 x3 x4 x5 x6) +. (#0.02)))`;;
9010 (* interval verification in partK.cc *)
9013 [((#4.0), x1, square_2t0);
9014 ((#4.0), x2, square_2t0);
9015 ((#4.0), x3, square_2t0);
9017 ((#8.0), x4, (#8.0));
9018 ((#4.0), x5, square_2t0);
9019 ((#4.0), x6, square_2t0)
9021 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.32))`;;
9026 (* interval verification in partK.cc *)
9028 CCC false in multiple branches of tauhat.
9029 Domain has been corrected. Should be flat quarters.
9031 CCC still false in vor0 branch.
9032 Not a counterexample, because the dihedral angle > 1.32.
9034 Bound: 0.0206833063205
9036 Point: [4.10991923445, 4.05029743735, 4.15049810846, 7.32673562767, 4.73630950763, 4.85438443725];
9038 yy = {4.10991923445, 4.05029743735, 4.15049810846, 7.32673562767, 4.73630950763, 4.85438443725}//Sqrt
9042 Dihedral @@ yy (* yields 1.651, so OK *)
9048 [((#4.0), x1, square_2t0);
9049 ((#4.0), x2, square_2t0);
9050 ((#4.0), x3, square_2t0);
9051 (square_2t0, x4, (#8.0));
9052 ((#4.0), x5, square_2t0);
9053 ((#4.0), x6, square_2t0)
9056 ( (tauhat_x x1 x2 x3 x4 x5 x6) >. ( (#3.07) *. pt)) \/
9057 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.32)))`;;
9061 (* interval verification in partK.cc *)
9065 ((#4.0), x1, square_2t0);
9066 ((#4.0), x2, square_2t0);
9067 ((#4.0), x3, square_2t0);
9068 (square_2t0, x4, #8.0);
9069 ((#4.0), x5, square_2t0);
9070 ((#4.0), x6, square_2t0)
9073 ((tau_0_x x1 x2 x3 x4 x5 x6) >. ((#3.07)*pt + xiV + (&2 * xi'_gamma))) \/
9074 ((dih_x x1 x2 x3 x4 x5 x6 >. (#1.32))) \/
9075 ((eta_x x4 x5 x6 <. sqrt2))
9079 let CIVA13_148776243= list_mk_conj
9080 [ I_705592875;I_747727191;I_474496219;I_649551700;I_74657942;
9081 I_897129160;I_760840103;I_675901554;I_712696695;I_269048407;
9082 I_553285469;I_293389410;I_695069283;I_814398901;I_352079526;
9087 LOC: 2002 IV, page 49
9091 (* interval verification by Ferguson *)
9092 (* let I_424011442= *)
9093 (* all_forall `ineq *)
9094 (* [((#4.0), x1, square_2t0); *)
9095 (* ((#4.0), x2, square_2t0); *)
9096 (* ((#4.0), x3, square_2t0); *)
9098 (* ((#4.0), x4, square_4t0); *)
9099 (* ((#4.0), x5, (square (#3.2))); *)
9100 (* (x5, x6, (square (#3.2))) *)
9103 (* ( (v0x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/ *)
9104 (* ( (sqrt x4) >. ( (sqrt x2) +. (sqrt x3))) \/ *)
9105 (* ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;; *)
9107 (* CCC made nonconstant bound a constraint *)
9110 [((#4.0), x1, square_2t0);
9111 ((#4.0), x2, square_2t0);
9112 ((#4.0), x3, square_2t0);
9114 ((#4.0), x4, square_4t0);
9115 ((#4.0), x5, (square (#3.2)));
9116 ((#4.0), x6, (square (#3.2)))
9119 ( (v0x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/
9120 ( (sqrt x4) >. ( (sqrt x2) +. (sqrt x3))) \/
9121 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/
9126 (* (\* interval verification by Ferguson *\) *)
9127 (* let I_140881233= *)
9128 (* all_forall `ineq *)
9129 (* [((#4.0), x1, square_2t0); *)
9130 (* ((#4.0), x2, square_2t0); *)
9131 (* ((#4.0), x3, square_2t0); *)
9133 (* ((#4.0), x4, square_4t0); *)
9134 (* ((#4.0), x5, (square (#3.2))); *)
9135 (* (x5, x6, (square (#3.2))) *)
9138 (* ( (v1x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/ *)
9139 (* ( (sqrt x4) >. ( (sqrt x2) +. (sqrt x3))) \/ *)
9140 (* ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;; *)
9144 (* interval verification by Ferguson *)
9145 (* CCC made nonconstant bound a constraint *)
9148 [((#4.0), x1, square_2t0);
9149 ((#4.0), x2, square_2t0);
9150 ((#4.0), x3, square_2t0);
9152 ((#4.0), x4, square_4t0);
9153 ((#4.0), x5, (square (#3.2)));
9154 ((#4.0), x6, (square (#3.2)))
9157 ( (v1x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/
9158 ( (sqrt x4) >. ( (sqrt x2) +. (sqrt x3))) \/
9159 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)) \/
9166 (* interval verification by Ferguson *)
9169 [((#4.0), x1, square_2t0);
9170 ((#4.0), x2, square_2t0);
9171 ((#4.0), x3, square_2t0);
9173 ((#8.0), x4, (square (#3.2)));
9174 ((#4.0), x5, (square (#2.189)));
9175 ((#4.0), x6, square_2t0)
9178 ( ( (v0x x1 x2 x3 x4 x5 x6) +. ( (#0.82) *. (sqrt (#421.0)))) <. (#0.0)) \/
9179 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
9184 (* interval verification by Ferguson *)
9187 [((#4.0), x1, square_2t0);
9188 ((#4.0), x2, square_2t0);
9189 ((#4.0), x3, square_2t0);
9191 ((#8.0), x4, (square (#3.2)));
9192 ((#4.0), x5, (square (#2.189)));
9193 ((#4.0), x6, square_2t0)
9196 ( ( (v1x x1 x2 x3 x4 x5 x6) +. ( (#0.82) *. (sqrt (#421.0)))) <. (#0.0)) \/
9197 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
9201 (* interval verification by Ferguson *)
9204 [((#4.0), x1, square_2t0);
9205 ((#4.0), x2, square_2t0);
9206 ((#4.0), x3, square_2t0);
9208 ((square (#3.2)), x4, square_4t0);
9209 ((#4.0), x5, (square (#2.189)));
9210 ((#4.0), x6, (square (#3.2)))
9213 ( ( (v0x x1 x2 x3 x4 x5 x6) +. ( (#0.82) *. (sqrt (#421.0)))) <. (#0.0)) \/
9214 ( (sqrt x4) >. ( (sqrt x2) +. (sqrt x3))) \/
9215 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
9219 (* interval verification by Ferguson *)
9222 [((#4.0), x1, square_2t0);
9223 ((#4.0), x2, square_2t0);
9224 ((#4.0), x3, square_2t0);
9226 ((square (#3.2)), x4, square_4t0);
9227 ((#4.0), x5, (square (#2.189)));
9228 ((#4.0), x6, (square (#3.2)))
9231 ( ( (v1x x1 x2 x3 x4 x5 x6) +. ( (#0.82) *. (sqrt (#421.0)))) <. (#0.0)) \/
9232 ( (sqrt x4) >. ( (sqrt x2) +. (sqrt x3))) \/
9233 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
9238 Two sets of bounds for x5 I used the more restrictive set
9240 (* interval verification by Ferguson *)
9243 [((#4.0), x1, square_2t0);
9244 ((#4.0), x2, square_2t0);
9245 ((#4.0), x3, square_2t0);
9247 ((#8.0), x4, (square (#3.2)));
9248 ((square (#2.189)), x5, square_2t0);
9249 ((#4.0), x6, square_2t0)
9252 ( ( (v0x x1 x2 x3 x4 x5 x6) +. ( (#0.5) *. (sqrt (#421.0)))) <. (#0.0)) \/
9253 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
9259 Two sets of bounds for x5 I used the more restrictive set
9261 (* interval verification by Ferguson *)
9264 [((#4.0), x1, square_2t0);
9265 ((#4.0), x2, square_2t0);
9266 ((#4.0), x3, square_2t0);
9268 ((#8.0), x4, (square (#3.2)));
9269 ((square (#2.189)), x5, square_2t0);
9270 ((#4.0), x6, square_2t0)
9273 ( ( (v1x x1 x2 x3 x4 x5 x6) +. ( (#0.5) *. (sqrt (#421.0)))) <. (#0.0)) \/
9274 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
9280 Two sets of bounds for x5 I used the more restrictive set
9283 (* interval verification by Ferguson *)
9286 [((#4.0), x1, square_2t0);
9287 ((#4.0), x2, square_2t0);
9288 ((#4.0), x3, square_2t0);
9290 ((square (#3.2)), x4, square_4t0);
9292 ((square (#2.189)), x5, (square (#3.2)));
9293 ((#4.0), x6, (square (#3.2)))
9296 ( ( (v0x x1 x2 x3 x4 x5 x6) +. ( (#0.5) *. (sqrt (#421.0)))) <. (#0.0)) \/
9297 ( (sqrt x4) >. ( (sqrt x2) +. (sqrt x3))) \/
9298 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
9304 Two sets of bounds for x5 I used the more restrictive set
9306 (* interval verification by Ferguson *)
9309 [((#4.0), x1, square_2t0);
9310 ((#4.0), x2, square_2t0);
9311 ((#4.0), x3, square_2t0);
9313 ((square (#3.2)), x4, square_4t0);
9315 ((square (#2.189)), x5, (square (#3.2)));
9316 ((#4.0), x6, (square (#3.2)))
9319 ( ( (v1x x1 x2 x3 x4 x5 x6) +. ( (#0.5) *. (sqrt (#421.0)))) <. (#0.0)) \/
9320 ( (sqrt x4) >. ( (sqrt x2) +. (sqrt x3))) \/
9321 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#0.0)))`;;
9325 (* interval verification by Ferguson *)
9328 [((#4.0), x1, square_2t0);
9329 ((#4.0), x2, square_2t0);
9330 ((#4.0), x3, square_2t0);
9332 ((#8.0), x4, square_4t0);
9333 ((#4.0), x5, (square (#3.2)));
9334 ((#4.0), x6, (square (#3.2)))
9337 ( (delta_x x1 x2 x3 x4 x5 x6) <. (#421.0)) \/
9338 ( (sqrt x4) >. ( (sqrt x2) +. (sqrt x3))) \/
9339 ( (eta_x x1 x3 x5) >. t0))`;;
9342 (* interval verification by Ferguson *)
9343 let I_484314425 = all_forall `ineq
9344 [((#4.0), x1, square_2t0);
9345 ((#4.0), x3, square_2t0);
9346 ((#4.0), x5, square_2t0)
9348 (--(&.4)*doct*(ups_x x1 x3 x5)*
9349 ((deriv (\x. (quo_x x1 x3 x)) x5) +.
9350 (deriv (\x. (quo_x x3 x1 x)) x5))
9353 (* interval verification by Ferguson *)
9354 let I_440223030 = all_forall `ineq
9355 [((#4.0), x1, square_2t0);
9356 ((#4.0), x3, square_2t0);
9357 ((square (#2.189)), x5, square_2t0)
9359 (--(&.4)*doct*(ups_x x1 x3 x5)*
9360 ((deriv (\x. (quo_x x1 x3 x)) x5) +.
9361 (deriv (\x. (quo_x x3 x1 x)) x5))
9365 Handwritten note says to change ( >=. ) to ( >. )
9366 overlap_f is the function of 1998:IV.4.11, or 2002,IV,Sec.4.14
9368 (* interval verification by Ferguson *)
9369 (* moved 115756648 to inequality_spec.ml *)
9373 let CIVA14_984628285 = list_mk_conj
9374 [ I_424011442;I_140881233;I_601456709_1;I_601456709_2;
9375 I_292977281_1;I_292977281_2;I_927286061_1;I_927286061_2;
9376 I_340409511_1;I_340409511_2;I_727498658;I_484314425;
9377 I_440223030;I_115756648;];;
9381 LOC: 2002 IV, page 49
9383 Remember to include this in the summary list-mk-conj
9386 (* interval verification by Ferguson *)
9387 let I_329882546_1= all_forall `ineq
9388 [((#4.0), x1, square_2t0);
9389 ((#4.0), x2, square_2t0);
9390 ((#4.0), x3, square_2t0);
9391 ((#8.0), x4, square_4t0);
9392 ((#4.0), x5, (#4.0));
9393 ((#4.0), x6, (#4.0))
9395 ((sqrt x4 >. (sqrt x2 + (sqrt x3))) \/
9396 (sqrt x4 >. (sqrt x5 + (sqrt x6))) \/
9397 (delta_x x1 x2 x3 x4 x5 x6 <. (&.0)) \/
9398 ~(deriv (\x. vor_0_x x x2 x3 x4 x5 x6) x1 = (&.0)) \/
9399 (deriv2 (\x. vor_0_x x x2 x3 x4 x5 x6) x1 >. (&.0)))`;;
9401 (* interval verification by Ferguson *)
9404 [((#4.0), x1, square_2t0);
9405 ((#4.0), x2, square_2t0);
9406 ((#4.0), x3, square_2t0);
9407 ((#8.0), x4, square_4t0);
9408 ((#4.0), x5, (#4.0));
9409 ((#4.0), x6, (#4.0))
9411 ((sqrt x4 >. (sqrt x2 + (sqrt x3))) \/
9412 (sqrt x4 >. (sqrt x5 + (sqrt x6))) \/
9413 (delta_x x1 x2 x3 x4 x5 x6 <. (&.0)) \/
9414 ~(deriv (\x. (-- (tau_0_x x x2 x3 x4 x5 x6))) x1 = (&.0)) \/
9415 (deriv2 (\x. (-- (tau_0_x x x2 x3 x4 x5 x6))) x1 >. (&.0)))`;;
9417 (* interval verification by Ferguson *)
9420 [((#4.0), x1, square_2t0);
9421 ((#4.0), x2, square_2t0);
9422 ((#4.0), x3, square_2t0);
9423 ((#8.0), x4, square_4t0);
9424 ((#4.0), x5, (#4.0));
9425 (square_2t0, x6, square_2t0)
9427 ((sqrt x4 >. (sqrt x2 + (sqrt x3))) \/
9428 (sqrt x4 >. (sqrt x5 + (sqrt x6))) \/
9429 (delta_x x1 x2 x3 x4 x5 x6 <. (&.0)) \/
9430 ~(deriv (\x. vor_0_x x x2 x3 x4 x5 x6) x1 = (&.0)) \/
9431 (deriv2 (\x. vor_0_x x x2 x3 x4 x5 x6) x1 >. (&.0)))`;;
9433 (* interval verification by Ferguson *)
9436 [((#4.0), x1, square_2t0);
9437 ((#4.0), x2, square_2t0);
9438 ((#4.0), x3, square_2t0);
9439 ((#8.0), x4, square_4t0);
9440 ((#4.0), x5, (#4.0));
9441 (square_2t0, x6, square_2t0)
9443 ((sqrt x4 >. (sqrt x2 + (sqrt x3))) \/
9444 (sqrt x4 >. (sqrt x5 + (sqrt x6))) \/
9445 (delta_x x1 x2 x3 x4 x5 x6 <. (&.0)) \/
9446 ~(deriv (\x. (-- (tau_0_x x x2 x3 x4 x5 x6))) x1 = (&.0)) \/
9447 (deriv2 (\x. (-- (tau_0_x x x2 x3 x4 x5 x6))) x1 >. (&.0)))`;;
9449 (* interval verification by Ferguson *)
9452 [((#4.0), x1, square_2t0);
9453 ((#4.0), x2, square_2t0);
9454 ((#4.0), x3, square_2t0);
9455 ((#8.0), x4, square_4t0);
9456 ((#4.0), x5, (#4.0));
9457 ((#8.0), x6, (#8.0))
9459 ((sqrt x4 >. (sqrt x2 + (sqrt x3))) \/
9460 (sqrt x4 >. (sqrt x5 + (sqrt x6))) \/
9461 (delta_x x1 x2 x3 x4 x5 x6 <. (&.0)) \/
9462 ~(deriv (\x. vor_0_x x x2 x3 x4 x5 x6) x1 = (&.0)) \/
9463 (deriv2 (\x. vor_0_x x x2 x3 x4 x5 x6) x1 >. (&.0)))`;;
9465 (* interval verification by Ferguson *)
9468 [((#4.0), x1, square_2t0);
9469 ((#4.0), x2, square_2t0);
9470 ((#4.0), x3, square_2t0);
9471 ((#8.0), x4, square_4t0);
9472 ((#4.0), x5, (#4.0));
9473 ((#8.0), x6, (#8.0))
9475 ((sqrt x4 >. (sqrt x2 + (sqrt x3))) \/
9476 (sqrt x4 >. (sqrt x5 + (sqrt x6))) \/
9477 (delta_x x1 x2 x3 x4 x5 x6 <. (&.0)) \/
9478 ~(deriv (\x. (-- (tau_0_x x x2 x3 x4 x5 x6))) x1 = (&.0)) \/
9479 (deriv2 (\x. (-- (tau_0_x x x2 x3 x4 x5 x6))) x1 >. (&.0)))`;;
9481 (* interval verification by Ferguson *)
9484 [((#4.0), x1, square_2t0);
9485 ((#4.0), x2, square_2t0);
9486 ((#4.0), x3, square_2t0);
9487 ((#8.0), x4, square_4t0);
9488 (square_2t0, x5, square_2t0);
9489 (square_2t0, x6, square_2t0)
9491 ((sqrt x4 >. (sqrt x2 + (sqrt x3))) \/
9492 (sqrt x4 >. (sqrt x5 + (sqrt x6))) \/
9493 (delta_x x1 x2 x3 x4 x5 x6 <. (&.0)) \/
9494 ~(deriv (\x. vor_0_x x x2 x3 x4 x5 x6) x1 = (&.0)) \/
9495 (deriv2 (\x. vor_0_x x x2 x3 x4 x5 x6) x1 >. (&.0)))`;;
9497 (* interval verification by Ferguson *)
9500 [((#4.0), x1, square_2t0);
9501 ((#4.0), x2, square_2t0);
9502 ((#4.0), x3, square_2t0);
9503 ((#8.0), x4, square_4t0);
9504 (square_2t0, x5, square_2t0);
9505 (square_2t0, x6, square_2t0)
9507 ((sqrt x4 >. (sqrt x2 + (sqrt x3))) \/
9508 (sqrt x4 >. (sqrt x5 + (sqrt x6))) \/
9509 (delta_x x1 x2 x3 x4 x5 x6 <. (&.0)) \/
9510 ~(deriv (\x. (-- (tau_0_x x x2 x3 x4 x5 x6))) x1 = (&.0)) \/
9511 (deriv2 (\x. (-- (tau_0_x x x2 x3 x4 x5 x6))) x1 >. (&.0)))`;;
9513 (* interval verification by Ferguson *)
9516 [((#4.0), x1, square_2t0);
9517 ((#4.0), x2, square_2t0);
9518 ((#4.0), x3, square_2t0);
9519 ((#8.0), x4, square_4t0);
9520 (square_2t0, x5, square_2t0);
9521 ((#8.0), x6, (#8.0))
9523 ((sqrt x4 >. (sqrt x2 + (sqrt x3))) \/
9524 (sqrt x4 >. (sqrt x5 + (sqrt x6))) \/
9525 (delta_x x1 x2 x3 x4 x5 x6 <. (&.0)) \/
9526 ~(deriv (\x. vor_0_x x x2 x3 x4 x5 x6) x1 = (&.0)) \/
9527 (deriv2 (\x. vor_0_x x x2 x3 x4 x5 x6) x1 >. (&.0)))`;;
9529 (* interval verification by Ferguson *)
9532 [((#4.0), x1, square_2t0);
9533 ((#4.0), x2, square_2t0);
9534 ((#4.0), x3, square_2t0);
9535 ((#8.0), x4, square_4t0);
9536 (square_2t0, x5, square_2t0);
9537 ((#8.0), x6, (#8.0))
9539 ((sqrt x4 >. (sqrt x2 + (sqrt x3))) \/
9540 (sqrt x4 >. (sqrt x5 + (sqrt x6))) \/
9541 (delta_x x1 x2 x3 x4 x5 x6 <. (&.0)) \/
9542 ~(deriv (\x. (-- (tau_0_x x x2 x3 x4 x5 x6))) x1 = (&.0)) \/
9543 (deriv2 (\x. (-- (tau_0_x x x2 x3 x4 x5 x6))) x1 >. (&.0)))`;;
9545 (* interval verification by Ferguson *)
9548 [((#4.0), x1, square_2t0);
9549 ((#4.0), x2, square_2t0);
9550 ((#4.0), x3, square_2t0);
9551 ((#8.0), x4, square_4t0);
9552 ((#8.0), x5, (#8.0));
9553 ((#8.0), x6, (#8.0))
9555 ((sqrt x4 >. (sqrt x2 + (sqrt x3))) \/
9556 (sqrt x4 >. (sqrt x5 + (sqrt x6))) \/
9557 (delta_x x1 x2 x3 x4 x5 x6 <. (&.0)) \/
9558 ~(deriv (\x. vor_0_x x x2 x3 x4 x5 x6) x1 = (&.0)) \/
9559 (deriv2 (\x. vor_0_x x x2 x3 x4 x5 x6) x1 >. (&.0)))`;;
9561 (* interval verification by Ferguson *)
9564 [((#4.0), x1, square_2t0);
9565 ((#4.0), x2, square_2t0);
9566 ((#4.0), x3, square_2t0);
9567 ((#8.0), x4, square_4t0);
9568 ((#8.0), x5, (#8.0));
9569 ((#8.0), x6, (#8.0))
9571 ((sqrt x4 >. (sqrt x2 + (sqrt x3))) \/
9572 (sqrt x4 >. (sqrt x5 + (sqrt x6))) \/
9573 (delta_x x1 x2 x3 x4 x5 x6 <. (&.0)) \/
9574 ~(deriv (\x. (-- (tau_0_x x x2 x3 x4 x5 x6))) x1 = (&.0)) \/
9575 (deriv2 (\x. (-- (tau_0_x x x2 x3 x4 x5 x6))) x1 >. (&.0)))`;;
9577 let CIVA15_311189443= list_mk_conj
9578 [ I_329882546_1;I_329882546_2;I_427688691_1;I_427688691_2;
9579 I_562103670_1;I_562103670_2;I_564506426_1;I_564506426_2;
9580 I_288224597_1;I_288224597_2;I_979916330_1;I_749968927_2;];;
9584 LOC: 2002 IV, page 49--50
9587 Comments from 2002 text:
9589 Some of these follow from known results.
9590 See II.4.5.1, F.3.13.1, F.3.13.3, F.3.13.4.
9592 The case vor <=0 of the inequality sigma<=0 for flat quarters
9593 follows by Rogers's monotonicity lemma I.8.6.2 and F.3.13.1,
9594 because the circumradius of the flat quarter is ASSUME_TAC least
9595 sqrt(2) when the analytic Voronoi function is used. We also
9596 use that vor(R(1,eta(2,2,2),sqrt(2)) = 0.
9603 ((#4.0), x1, square_2t0);
9604 ((#4.0), x2, square_2t0);
9605 ((#4.0), x3, square_2t0);
9606 (square_2t0, x4, (#8.0));
9607 ((#4.0), x5, square_2t0);
9608 ((#4.0), x6, square_2t0)
9610 (taumu_flat_x x1 x2 x3 x4 x5 x6 >. #0.06585)`;;
9615 ((square (#2.2)), x1, square_2t0);
9616 ((#4.0), x2, square_2t0);
9617 ((#4.0), x3, square_2t0);
9618 ((square(#2.6)), x4, (#8.0));
9619 ((#4.0), x5, square_2t0);
9620 ((#4.0), x6, square_2t0)
9622 ((#0.0063) + (tau_0_x x1 x2 x3 x4 x5 x6) >. #0.06585)`;;
9627 ((#4.0), x1, (square (#2.2)));
9628 ((#4.0), x2, square_2t0);
9629 ((#4.0), x3, square_2t0);
9630 ((square(#2.7)), x4, (#8.0));
9631 ((#4.0), x5, square_2t0);
9632 ((#4.0), x6, square_2t0)
9634 ((#0.0114) + (tau_0_x x1 x2 x3 x4 x5 x6) >. #0.06585)`;;
9636 (* In this fourth case, we get half from each upright quarter. *)
9640 (square_2t0, x1, (#8.0));
9641 ((#4.0), x2, square_2t0);
9642 ((#4.0), x3, square_2t0);
9643 ((#4.0), x4, square_2t0);
9644 ((#4.0), x5, square_2t0);
9645 ((#4.0), x6, square_2t0)
9647 ((taunu_x x1 x2 x3 x4 x5 x6) >. (#0.06585)/(#2.0))`;;
9652 ((#4.0), x1, square_2t0);
9653 ((square(#2.23)), x2, square_2t0);
9654 ((#4.0), x3, square_2t0);
9655 ((square(#2.77)), x4, (#8.0));
9656 ((#4.0), x5, square_2t0);
9657 ((#4.0), x6, square_2t0)
9659 (((tau_0_x x1 x2 x3 x4 x5 x6) >. #0.06585) \/
9660 (eta_x x4 x5 x6 <. (sqrt (#2.0))))`;;
9662 (* direction of inequality corrected in 690626704_* on Dec 16, 2007, tch *)
9667 ((#4.0), x1, square_2t0);
9668 ((#4.0), x2, square_2t0);
9669 ((#4.0), x3, square_2t0);
9670 (square_2t0, x4, (#8.0));
9671 ((#4.0), x5, square_2t0);
9672 ((#4.0), x6, square_2t0)
9674 (mu_flat_x x1 x2 x3 x4 x5 x6 <. #0.0)`;;
9679 ((square (#2.2)), x1, square_2t0);
9680 ((#4.0), x2, square_2t0);
9681 ((#4.0), x3, square_2t0);
9682 ((square(#2.6)), x4, (#8.0));
9683 ((#4.0), x5, square_2t0);
9684 ((#4.0), x6, square_2t0)
9686 ((--(#0.0063)) + (vor_0_x x1 x2 x3 x4 x5 x6) <. #0.0)`;;
9691 ((#4.0), x1, (square (#2.2)));
9692 ((#4.0), x2, square_2t0);
9693 ((#4.0), x3, square_2t0);
9694 ((square(#2.7)), x4, (#8.0));
9695 ((#4.0), x5, square_2t0);
9696 ((#4.0), x6, square_2t0)
9698 ((--(#0.0114)) + (vor_0_x x1 x2 x3 x4 x5 x6) <. #0.0)`;;
9700 (* In this fourth case, we get half from each upright quarter. *)
9704 (square_2t0, x1, (#8.0));
9705 ((#4.0), x2, square_2t0);
9706 ((#4.0), x3, square_2t0);
9707 ((#4.0), x4, square_2t0);
9708 ((#4.0), x5, square_2t0);
9709 ((#4.0), x6, square_2t0)
9711 ((nu_x x1 x2 x3 x4 x5 x6) <. (#0.0))`;;
9716 ((#4.0), x1, square_2t0);
9717 ((square(#2.23)), x2, square_2t0);
9718 ((#4.0), x3, square_2t0);
9719 ((square(#2.77)), x4, (#8.0));
9720 ((#4.0), x5, square_2t0);
9721 ((#4.0), x6, square_2t0)
9723 (((vor_0_x x1 x2 x3 x4 x5 x6) <. #0.0) \/
9724 (eta_x x4 x5 x6 <. (sqrt (#2.0))))`;;
9729 [((#4.0), x1, square_2t0);
9730 ((#4.0), x2, square_2t0);
9731 ((#4.0), x3, square_2t0);
9733 (square_2t0, x4, (square (#2.77)));
9734 (square_2t0, x5, (square (#2.77)));
9735 ((#4.0), x6, square_2t0)
9738 ( (vor_analytic_x x1 x2 x3 x4 x5 x6) <. (--. (#0.05714))) \/
9739 ( (eta_x x4 x5 x6) >. (sqrt (#2.0))))`;;
9744 [((#4.0), x1, square_2t0);
9745 ((#4.0), x2, square_2t0);
9746 ((#4.0), x3, square_2t0);
9748 (square_2t0, x4, (square (#2.77)));
9749 (square_2t0, x5, (square (#2.77)));
9750 ((#4.0), x6, square_2t0)
9753 ( (tau_analytic_x x1 x2 x3 x4 x5 x6) >. (#0.13943)) \/
9754 ( (eta_x x4 x5 x6) >. (sqrt (#2.0))))`;;
9756 (* STM 1/13/08. Added parentheses. This was not parsing correctly *)
9758 CCC false. Sign of the inequality corrected on the eta constraint 1/31/2008.
9760 Bound: 0.0133663042564
9762 Point: [3.99999999999, 3.99999999999, 3.99999999999, 3.99999999999, 6.30009999999, 6.30009999999]
9767 [((#4.0), x1, square_2t0);
9768 ((#4.0), x2, square_2t0);
9769 ((#4.0), x3, square_2t0);
9770 ((#4.0), x4, square_2t0);
9771 (square_2t0, x5, (#8.0));
9772 (square_2t0, x6, (#8.0))
9774 ((vor_0_x x1 x2 x3 x4 x5 x6 <. Z32) \/
9775 (eta_x x4 x5 x6 <. (sqrt (#2.0)) ))`;;
9779 [((#4.0), x1, square_2t0);
9780 ((#4.0), x2, square_2t0);
9781 ((#4.0), x3, square_2t0);
9782 ((#4.0), x4, square_2t0);
9783 ((square(#2.77), x5, (#8.0)));
9784 (square_2t0, x6, (#8.0))
9786 (vor_0_x x1 x2 x3 x4 x5 x6 <. Z32)`;;
9788 (* STM 1/13/08. Added parentheses. This was not parsing correctly *)
9790 CCC false. Sign of the inequality corrected on the eta constraint 1/31/2008.
9792 Bound: 0.0130374551969
9794 Point: [3.99999999999, 3.99999999999, 3.99999999999, 3.99999999999, 6.30009999999, 6.30009999999]
9799 [((#4.0), x1, square_2t0);
9800 ((#4.0), x2, square_2t0);
9801 ((#4.0), x3, square_2t0);
9802 ((#4.0), x4, square_2t0);
9803 (square_2t0, x5, (#8.0));
9804 (square_2t0, x6, (#8.0))
9806 ((tau_0_x x1 x2 x3 x4 x5 x6 >. (#0.13943)) \/
9807 (eta_x x4 x5 x6 <. (sqrt (#2.0)) ))`;;
9811 [((#4.0), x1, square_2t0);
9812 ((#4.0), x2, square_2t0);
9813 ((#4.0), x3, square_2t0);
9814 ((#4.0), x4, square_2t0);
9815 ((square(#2.77), x5, (#8.0)));
9816 (square_2t0, x6, (#8.0))
9818 (tau_0_x x1 x2 x3 x4 x5 x6 >. (#0.13943))`;;
9820 let CIVA16_163548682 = list_mk_conj
9821 [ I_695180203_1;I_695180203_2;I_695180203_3;I_695180203_4;
9822 I_695180203_5;I_690626704_1;I_690626704_2;I_690626704_3;
9823 I_690626704_4;I_690626704_5;I_807023313;I_590577214;
9824 I_949210508_1;I_949210508_2;I_671961774_1;I_671961774_2;];;
9828 LOC: 2002 IV, page 50
9849 (* interval verification by Ferguson *)
9850 let I_645264496_102=
9852 [((#4.0), x1, square_2t0);
9853 ((#4.0), x2, square_2t0);
9854 ((#4.0), x3, square_2t0);
9855 ((#4.0), x4, square_2t0);
9856 ((#8.0), x5, (square (#3.2)));
9857 ((#8.0), x6, (square (#3.2)))
9859 ((tau_0_x x1 x2 x3 x4 x5 x6)- (pi_prime_tau 1 0 2) >. D32)`;;
9861 (* interval verification by Ferguson *)
9862 let I_645264496_111=
9864 [((#4.0), x1, square_2t0);
9865 ((#4.0), x2, square_2t0);
9866 ((#4.0), x3, square_2t0);
9867 ((#4.0), x4, square_2t0);
9868 (square_2t0, x5, (#8.0));
9869 ((#8.0), x6, (square (#3.2)))
9871 ((tau_0_x x1 x2 x3 x4 x5 x6)- (pi_prime_tau 1 1 1) >. D32)`;;
9873 (* interval verification by Ferguson *)
9874 let I_645264496_030=
9876 [((#4.0), x1, square_2t0);
9877 ((#4.0), x2, square_2t0);
9878 ((#4.0), x3, square_2t0);
9879 (square_2t0, x4, (#8.0));
9880 (square_2t0, x5, (#8.0));
9881 (square_2t0, x6, (#8.0))
9883 ((tau_0_x x1 x2 x3 x4 x5 x6)- (pi_prime_tau 0 3 0) >. D33)`;;
9885 (* interval verification by Ferguson *)
9886 let I_645264496_021=
9888 [((#4.0), x1, square_2t0);
9889 ((#4.0), x2, square_2t0);
9890 ((#4.0), x3, square_2t0);
9891 (square_2t0, x4, (#8.0));
9892 (square_2t0, x5, (#8.0));
9893 ((#8.0), x6, (square (#3.2)))
9895 ((tau_0_x x1 x2 x3 x4 x5 x6)- (pi_prime_tau 0 2 1) >. D33)`;;
9897 (* interval verification by Ferguson *)
9898 let I_645264496_012=
9900 [((#4.0), x1, square_2t0);
9901 ((#4.0), x2, square_2t0);
9902 ((#4.0), x3, square_2t0);
9903 (square_2t0, x4, (#8.0));
9904 ((#8.0), x5, (square (#3.2)));
9905 ((#8.0), x6, (square (#3.2)))
9907 ((tau_0_x x1 x2 x3 x4 x5 x6)- (pi_prime_tau 0 1 2) >. D33)`;;
9909 (* interval verification by Ferguson *)
9910 let I_645264496_003=
9912 [((#4.0), x1, square_2t0);
9913 ((#4.0), x2, square_2t0);
9914 ((#4.0), x3, square_2t0);
9915 ((#8.0), x4, (square (#3.2)));
9916 ((#8.0), x5, (square (#3.2)));
9917 ((#8.0), x6, (square (#3.2)))
9919 ((tau_0_x x1 x2 x3 x4 x5 x6)- (pi_prime_tau 0 0 3) >. D33)`;;
9924 (* interval verification by Ferguson *)
9927 [((#4.0), x1, square_2t0);
9928 ((#4.0), x2, square_2t0);
9929 ((#4.0), x3, square_2t0);
9931 ((square (#2.6)), x4, (#8.0));
9932 ((#8.0), x5, (square (#3.2)));
9933 ((#4.0), x6, square_2t0)
9935 ( ( (tau_0_x x1 x2 x3 x4 x5 x6) +. (--. (#0.034052))) >. (#0.13943))`;;
9939 (* interval verification by Ferguson *)
9942 [((#4.0), x1, square_2t0);
9943 ((#4.0), x2, square_2t0);
9944 ((#4.0), x3, square_2t0);
9946 (square_2t0, x4, square_2t0);
9947 ((square (#3.2)), x5, (square (#3.2)));
9948 ((#4.0), x6, (#4.0))
9950 ( ( (tau_0_x x1 x2 x3 x4 x5 x6) +. (--. (#0.034052)) +. (--. (#0.0066))) >. (#0.13943))`;;
9953 let CIVA17_852270725 = list_mk_conj
9954 [ I_645264496_102;I_645264496_111;I_645264496_030;I_645264496_021;
9955 I_645264496_012;I_645264496_003;I_910154674;I_877743345;];;
9959 LOC: 2002 IV, page 50
9965 (* interval verification by Ferguson *)
9966 let I_612259047_102=
9968 [((#4.0), x1, square_2t0);
9969 ((#4.0), x2, square_2t0);
9970 ((#4.0), x3, square_2t0);
9971 ((#4.0), x4, square_2t0);
9972 ((#8.0), x5, (square (#3.2)));
9973 ((#8.0), x6, (square (#3.2)))
9975 ((vor_0_x x1 x2 x3 x4 x5 x6)+ (pi_prime_sigma 1 0 2) <. Z32)`;;
9977 (* interval verification by Ferguson *)
9978 let I_612259047_111=
9980 [((#4.0), x1, square_2t0);
9981 ((#4.0), x2, square_2t0);
9982 ((#4.0), x3, square_2t0);
9983 ((#4.0), x4, square_2t0);
9984 (square_2t0, x5, (#8.0));
9985 ((#8.0), x6, (square (#3.2)))
9987 ((vor_0_x x1 x2 x3 x4 x5 x6)+ (pi_prime_sigma 1 1 1) <. Z32)`;;
9989 (* interval verification by Ferguson *)
9990 let I_612259047_030=
9992 [((#4.0), x1, square_2t0);
9993 ((#4.0), x2, square_2t0);
9994 ((#4.0), x3, square_2t0);
9995 (square_2t0, x4, (#8.0));
9996 (square_2t0, x5, (#8.0));
9997 (square_2t0, x6, (#8.0))
9999 ((vor_0_x x1 x2 x3 x4 x5 x6)+ (pi_prime_sigma 0 3 0) <. Z33)`;;
10001 (* interval verification by Ferguson *)
10002 let I_612259047_021=
10004 [((#4.0), x1, square_2t0);
10005 ((#4.0), x2, square_2t0);
10006 ((#4.0), x3, square_2t0);
10007 (square_2t0, x4, (#8.0));
10008 (square_2t0, x5, (#8.0));
10009 ((#8.0), x6, (square (#3.2)))
10011 ((vor_0_x x1 x2 x3 x4 x5 x6)+ (pi_prime_sigma 0 2 1) <. Z33)`;;
10013 (* interval verification by Ferguson *)
10014 let I_612259047_012=
10016 [((#4.0), x1, square_2t0);
10017 ((#4.0), x2, square_2t0);
10018 ((#4.0), x3, square_2t0);
10019 (square_2t0, x4, (#8.0));
10020 ((#8.0), x5, (square (#3.2)));
10021 ((#8.0), x6, (square (#3.2)))
10023 ((vor_0_x x1 x2 x3 x4 x5 x6)+ (pi_prime_sigma 0 1 2) <. Z33)`;;
10025 (* interval verification by Ferguson *)
10026 let I_612259047_003=
10028 [((#4.0), x1, square_2t0);
10029 ((#4.0), x2, square_2t0);
10030 ((#4.0), x3, square_2t0);
10031 ((#8.0), x4, (square (#3.2)));
10032 ((#8.0), x5, (square (#3.2)));
10033 ((#8.0), x6, (square (#3.2)))
10035 ((vor_0_x x1 x2 x3 x4 x5 x6)+ (pi_prime_sigma 0 0 3) <. Z33)`;;
10038 let CIVA18_819209129 = list_mk_conj
10039 [ I_612259047_102;I_612259047_111;I_612259047_030;I_612259047_021;
10040 I_612259047_012;I_612259047_003;];;
10044 LOC: 2002 IV, page 50
10047 Note: I might need to add some convexity results to make what
10048 is stated below consistent with what is asserted in 2002-IV.
10050 Without loss of generality in Section 19, we can divide the
10051 quad along the shorter diagonal.
10054 (* interval verification by Ferguson *)
10057 [((#4.0), x1, square_2t0);
10058 ((#4.0), x3, square_2t0);
10059 ((#8.0), x4, (square (#3.2)));
10060 (square_2t0, x5, (#8.0))
10062 (((tau_0_x x1 (#4.0) x3 x4 x5 (#4.0))+
10063 (tau_0_x (#4.0) (#4.0) x3 x4 (#4.0) (#4.0)) >. (#0.235)) \/
10064 (cross_diag_x x1 (#4.0) x3 x4 x5 (#4.0) (#4.0) (#4.0) (#4.0)
10067 (* interval verification by Ferguson *)
10070 [((#4.0), x1, square_2t0);
10071 ((#4.0), x3, square_2t0);
10072 ((#8.0), x4, (square (#3.2)));
10073 ((#8.0), x5, (square (#3.2)))
10075 (((tau_0_x x1 (#4.0) x3 x4 x5 (#4.0))+
10076 (tau_0_x (#4.0) (#4.0) x3 x4 (#4.0) (#4.0)) >. (#0.3109)) \/
10077 (cross_diag_x x1 (#4.0) x3 x4 x5 (#4.0) (#4.0) (#4.0) (#4.0)
10080 (* interval verification by Ferguson *)
10083 [((#4.0), x1, square_2t0);
10084 ((#4.0), x3, square_2t0);
10085 ((#8.0), x4, (square (#3.2)));
10086 (square_2t0, x5, (#8.0))
10088 (((vor_0_x x1 (#4.0) x3 x4 x5 (#4.0))+
10089 (vor_0_x (#4.0) (#4.0) x3 x4 (#4.0) (#4.0)) <. (--(#0.075))) \/
10090 (cross_diag_x x1 (#4.0) x3 x4 x5 (#4.0) (#4.0) (#4.0) (#4.0)
10093 (* interval verification by Ferguson *)
10096 [((#4.0), x1, square_2t0);
10097 ((#4.0), x3, square_2t0);
10098 ((#8.0), x4, (square (#3.2)));
10099 ((#8.0), x5, (square (#3.2)))
10101 (((vor_0_x x1 (#4.0) x3 x4 x5 (#4.0))+
10102 (vor_0_x (#4.0) (#4.0) x3 x4 (#4.0) (#4.0)) <. (--(#0.137))) \/
10103 (cross_diag_x x1 (#4.0) x3 x4 x5 (#4.0) (#4.0) (#4.0) (#4.0)
10106 let CIVA19_128523606 = list_mk_conj
10107 [ I_357477295_1;I_357477295_2;I_357477295_3;I_357477295_4;];;
10111 LOC: 2002 IV, page 50--51
10114 Let $Q$ be a quadrilateral subcluster
10115 whose edges are described by the vector
10116 $$(2,2,a_2,2,2,b_3,a_4,b_4).$$
10117 Assume $b_4\ge b_3$, $b_4\in\{2t_0,2\sqrt2\}$,
10118 $b_3\in\{2,2t_0,2\sqrt2\}$, $a_2,a_4\in\{2,2t_0\}$. Assume that the
10119 diagonal between corners $1$ and $3$ has length in $[2\sqrt2,3.2]$, and
10120 that the other diagonal has length $\ge3.2$. Let $k_0$, $k_1$, $k_2$ be
10121 the number of $b_i$ equal to $2$, $2t_0$, $2\sqrt2$, respectively. If
10122 $b_4=2t_0$ and $b_3=2$, no such subcluster exists (the reader can check
10123 that $\Delta(4,4,x_3,4,2t_0^2,x_6)<0$ under these conditions), and we
10138 (* interval verification by Ferguson *)
10139 let I_193776341_GEN=
10140 `(\ b4 b3 a2 a4 k0 k1 k2. (
10148 ((#8.0), (x4:real), (square (#3.2)))]
10149 (((vor_0_x x1 x2 x3 x4 x5 x6) +
10150 (vor_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) <.
10151 ((if (k1+k2 = 2) then Z42 else Z41) -
10152 ((#0.009)*(&.k2) + (&. (k0+ 2 *k2))*((#0.008)/(#3.0))))
10155 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10156 <. ((#3.2)))))))`;;
10158 (* interval verification by Ferguson *)
10160 all_forall (list_mk_comb(I_193776341_GEN,
10161 [`#8.0`;`#8.0`;`#4.0`;`#4.0`;`0`;`0`;`2`]));;
10163 (* interval verification by Ferguson *)
10165 all_forall (list_mk_comb(I_193776341_GEN,
10166 [`#8.0`;`#8.0`;`#4.0`;`square_2t0`;`0`;`0`;`2`]));;
10168 (* interval verification by Ferguson *)
10170 all_forall (list_mk_comb(I_193776341_GEN,
10171 [`#8.0`;`#8.0`;`square_2t0`;`square_2t0`;`0`;`0`;`2`]));;
10173 (* interval verification by Ferguson *)
10175 all_forall (list_mk_comb(I_193776341_GEN,
10176 [`#8.0`;`#8.0`;`square_2t0`;`#4.0`;`0`;`0`;`2`]));;
10178 (* interval verification by Ferguson *)
10180 all_forall (list_mk_comb(I_193776341_GEN,
10181 [`#8.0`;`square_2t0`;`#4.0`;`#4.0`;`0`;`1`;`1`]));;
10183 (* interval verification by Ferguson *)
10185 all_forall (list_mk_comb(I_193776341_GEN,
10186 [`#8.0`;`square_2t0`;`#4.0`;`square_2t0`;`0`;`1`;`1`]));;
10188 (* interval verification by Ferguson *)
10190 all_forall (list_mk_comb(I_193776341_GEN,
10191 [`#8.0`;`square_2t0`;`square_2t0`;`square_2t0`;`0`;`1`;`1`]));;
10193 (* interval verification by Ferguson *)
10195 all_forall (list_mk_comb(I_193776341_GEN,
10196 [`#8.0`;`square_2t0`;`square_2t0`;`#4.0`;`0`;`1`;`1`]));;
10198 (* interval verification by Ferguson *)
10200 all_forall (list_mk_comb(I_193776341_GEN,
10201 [`#8.0`;`#4.0`;`#4.0`;`#4.0`;`1`;`0`;`1`]));;
10203 (* interval verification by Ferguson *)
10204 let I_193776341_10=
10205 all_forall (list_mk_comb(I_193776341_GEN,
10206 [`#8.0`;`#4.0`;`#4.0`;`square_2t0`;`1`;`0`;`1`]));;
10208 (* interval verification by Ferguson *)
10209 let I_193776341_11=
10210 all_forall (list_mk_comb(I_193776341_GEN,
10211 [`#8.0`;`#4.0`;`square_2t0`;`square_2t0`;`1`;`0`;`1`]));;
10213 (* interval verification by Ferguson *)
10214 let I_193776341_12=
10215 all_forall (list_mk_comb(I_193776341_GEN,
10216 [`#8.0`;`#4.0`;`square_2t0`;`#4.0`;`1`;`0`;`1`]));;
10218 (* interval verification by Ferguson *)
10219 let I_193776341_13= all_forall (list_mk_comb(I_193776341_GEN,
10220 [`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`0`;`2`;`0`]));;
10222 (* interval verification by Ferguson *)
10223 let I_193776341_14=
10224 all_forall (list_mk_comb(I_193776341_GEN,
10225 [`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`0`;`2`;`0`]));;
10227 (* interval verification by Ferguson *)
10228 let I_193776341_15=
10229 all_forall (list_mk_comb(I_193776341_GEN,
10230 [`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`0`;`2`;`0`]));;
10232 (* interval verification by Ferguson *)
10233 let I_193776341_16=
10234 all_forall (list_mk_comb(I_193776341_GEN,
10235 [`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`0`;`2`;`0`]));;
10237 (* interval verification by Ferguson *)
10238 let I_898647773_GEN=
10239 `(\ b4 b3 a2 a4 k0 k1 k2. (
10247 ((#8.0), (x4:real), (square (#3.2)))]
10248 (((tau_0_x x1 x2 x3 x4 x5 x6) +
10249 (tau_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) >.
10250 ((if (k1+k2 = 2) then D42 else D41) + (#0.04683) +
10251 ((#0.0066)*(&.k2) + ((&. (k0+ 2 *k2))-(#3.0))*((#0.008)/(#3.0))))
10254 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10255 <. ((#3.2)))))))`;;
10258 (* interval verification by Ferguson *)
10260 all_forall (list_mk_comb(I_898647773_GEN,
10261 [`#8.0`;`#8.0`;`#4.0`;`#4.0`;`0`;`0`;`2`]));;
10263 (* interval verification by Ferguson *)
10265 all_forall (list_mk_comb(I_898647773_GEN,
10266 [`#8.0`;`#8.0`;`#4.0`;`square_2t0`;`0`;`0`;`2`]));;
10268 (* interval verification by Ferguson *)
10270 all_forall (list_mk_comb(I_898647773_GEN,
10271 [`#8.0`;`#8.0`;`square_2t0`;`square_2t0`;`0`;`0`;`2`]));;
10273 (* interval verification by Ferguson *)
10275 all_forall (list_mk_comb(I_898647773_GEN,
10276 [`#8.0`;`#8.0`;`square_2t0`;`#4.0`;`0`;`0`;`2`]));;
10278 (* interval verification by Ferguson *)
10280 all_forall (list_mk_comb(I_898647773_GEN,
10281 [`#8.0`;`square_2t0`;`#4.0`;`#4.0`;`0`;`1`;`1`]));;
10283 (* interval verification by Ferguson *)
10284 let I_898647773_6= all_forall (list_mk_comb(I_898647773_GEN,
10285 [`#8.0`;`square_2t0`;`#4.0`;`square_2t0`;`0`;`1`;`1`]));;
10287 (* interval verification by Ferguson *)
10289 all_forall (list_mk_comb(I_898647773_GEN,
10290 [`#8.0`;`square_2t0`;`square_2t0`;`square_2t0`;`0`;`1`;`1`]));;
10292 (* interval verification by Ferguson *)
10294 all_forall (list_mk_comb(I_898647773_GEN,
10295 [`#8.0`;`square_2t0`;`square_2t0`;`#4.0`;`0`;`1`;`1`]));;
10297 (* interval verification by Ferguson *)
10299 all_forall (list_mk_comb(I_898647773_GEN,
10300 [`#8.0`;`#4.0`;`#4.0`;`#4.0`;`1`;`0`;`1`]));;
10302 (* interval verification by Ferguson *)
10303 let I_898647773_10=
10304 all_forall (list_mk_comb(I_898647773_GEN,
10305 [`#8.0`;`#4.0`;`#4.0`;`square_2t0`;`1`;`0`;`1`]));;
10307 (* interval verification by Ferguson *)
10308 let I_898647773_11=
10309 all_forall (list_mk_comb(I_898647773_GEN,
10310 [`#8.0`;`#4.0`;`square_2t0`;`square_2t0`;`1`;`0`;`1`]));;
10312 (* interval verification by Ferguson *)
10313 let I_898647773_12=
10314 all_forall (list_mk_comb(I_898647773_GEN,
10315 [`#8.0`;`#4.0`;`square_2t0`;`#4.0`;`1`;`0`;`1`]));;
10317 (* interval verification by Ferguson *)
10318 let I_898647773_13=
10319 all_forall (list_mk_comb(I_898647773_GEN,
10320 [`square_2t0`;`square_2t0`;`#4.0`;`#4.0`;`0`;`2`;`0`]));;
10322 (* interval verification by Ferguson *)
10323 let I_898647773_14=
10324 all_forall (list_mk_comb(I_898647773_GEN,
10325 [`square_2t0`;`square_2t0`;`#4.0`;`square_2t0`;`0`;`2`;`0`]));;
10327 (* interval verification by Ferguson *)
10328 let I_898647773_15=
10329 all_forall (list_mk_comb(I_898647773_GEN,
10330 [`square_2t0`;`square_2t0`;`square_2t0`;`square_2t0`;`0`;`2`;`0`]));;
10332 (* interval verification by Ferguson *)
10333 let I_898647773_16=
10334 all_forall (list_mk_comb(I_898647773_GEN,
10335 [`square_2t0`;`square_2t0`;`square_2t0`;`#4.0`;`0`;`2`;`0`]));;
10337 (* STM 1/13/08. Added parentheses. This was not parsing correctly *)
10338 (* interval verification by Ferguson *)
10352 ((#8.0), (x4:real), (square (#3.2)))]
10353 ((((vor_0_x x1 x2 x3 x4 x5 x6) +
10354 (vor_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) <.
10355 Z42 - (#0.0461) - (#0.009) - (&.2)*(#0.008)))
10357 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10360 (* STM 1/13/08. Added parentheses. This was not parsing correctly *)
10361 (* interval verification by Ferguson *)
10364 let a2 = (square_2t0) in
10375 ((#8.0), (x4:real), (square (#3.2)))]
10376 ((((vor_0_x x1 x2 x3 x4 x5 x6) +
10377 (vor_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) <.
10378 Z42 - (#0.0461) - (#0.009) - (&.2)*(#0.008)))
10380 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10383 (* STM 1/13/08. Added parentheses. This was not parsing correctly *)
10384 (* interval verification by Ferguson *)
10390 let b3 = (square_2t0) in
10398 ((#8.0), (x4:real), (square (#3.2)))]
10399 ((((vor_0_x x1 x2 x3 x4 x5 x6) +
10400 (vor_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) <.
10401 Z42 - (#0.0461) - (#0.009) - (&.2)*(#0.008)))
10403 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10406 (* STM 1/13/08. Added parentheses. This was not parsing correctly *)
10407 (* interval verification by Ferguson *)
10410 let a2 = (square_2t0) in
10413 let b3 = (square_2t0) in
10420 [((#8.0), (x4:real), (square (#3.2)))]
10421 ((((vor_0_x x1 x2 x3 x4 x5 x6) +
10422 (vor_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) <.
10423 (Z42 - (#0.0461) - (#0.009) - (&.2)*(#0.008))))
10425 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10429 (* interval verification by Ferguson *)
10443 ((#8.0), (x4:real), (square (#3.2)))]
10444 ((((tau_0_x x1 x2 x3 x4 x5 x6) +
10445 (tau_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) >.
10446 D51 + (#0.04683)+(#0.008)+(&.2)*(#0.066)))
10448 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10451 (* interval verification by Ferguson *)
10454 let a2 = (square_2t0) in
10465 ((#8.0), (x4:real), (square (#3.2)))]
10466 ((((tau_0_x x1 x2 x3 x4 x5 x6) +
10467 (tau_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) >.
10468 D51 + (#0.04683)+(#0.008)+(&.2)*(#0.066)))
10470 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10473 (* STM 1/13/08. Added parentheses. This was not parsing correctly *)
10474 (* interval verification by Ferguson *)
10480 let b3 = (square_2t0) in
10488 ((#8.0), (x4:real), (square (#3.2)))]
10489 ((((tau_0_x x1 x2 x3 x4 x5 x6) +
10490 (tau_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) >.
10491 D51 + (#0.04683)+(#0.008)+(&.2)*(#0.066)))
10493 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10496 (* STM 1/13/08. Added parentheses. This was not parsing correctly *)
10497 (* interval verification by Ferguson *)
10500 let a2 = (square_2t0) in
10503 let b3 = (square_2t0) in
10510 [((#8.0), (x4:real), (square (#3.2)))]
10511 ((((tau_0_x x1 x2 x3 x4 x5 x6) +
10512 (tau_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) >.
10513 D51 + (#0.04683)+(#0.008)+(&.2)*(#0.066)))
10515 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10519 (* interval verification by Ferguson *)
10533 ((#8.0), (x4:real), (square (#3.2)))]
10534 (((vor_0_x x1 x2 x3 x4 x5 x6) +
10535 (vor_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) <.
10536 s5 - (#0.0461) - (#0.008))
10538 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10541 (* interval verification by Ferguson *)
10544 let a2 = (square_2t0) in
10555 ((#8.0), (x4:real), (square (#3.2)))]
10556 (((vor_0_x x1 x2 x3 x4 x5 x6) +
10557 (vor_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) <.
10558 s5 - (#0.0461) - (#0.008))
10560 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10563 (* interval verification by Ferguson *)
10577 ((#8.0), (x4:real), (square (#3.2)))]
10578 (((tau_0_x x1 x2 x3 x4 x5 x6) +
10579 (tau_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) >.
10582 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10585 (* interval verification by Ferguson *)
10588 let a2 = (square_2t0) in
10599 ((#8.0), (x4:real), (square (#3.2)))]
10600 (((tau_0_x x1 x2 x3 x4 x5 x6) +
10601 (tau_0_x (a2) x2 x3 x4 (#4.0) (#4.0)) >.
10604 (cross_diag_x x1 x2 x3 x4 x5 x6 a2 (#4.0) (#4.0)
10607 let CIVA20_874876755 = list_mk_conj
10608 [ I_193776341_1;I_193776341_2;I_193776341_3;I_193776341_4;
10609 I_193776341_5;I_193776341_6;I_193776341_7;I_193776341_8;
10610 I_193776341_9;I_193776341_10;I_193776341_11;I_193776341_12;
10611 I_193776341_13;I_193776341_14;I_193776341_15;I_193776341_16;
10612 I_898647773_1;I_898647773_2;I_898647773_3;I_898647773_4;
10613 I_898647773_5;I_898647773_6;I_898647773_7;I_898647773_8;
10614 I_898647773_9;I_898647773_10;I_898647773_11;I_898647773_12;
10615 I_898647773_13;I_898647773_14;I_898647773_15;I_898647773_16;
10616 I_844634710_1;I_844634710_2;I_844634710_3;I_844634710_4;
10617 I_328845176_1;I_328845176_2;I_328845176_3;I_328845176_4;
10618 I_233273785_1;I_233273785_2;I_96695550_1;I_96695550_2;];;
10621 LOC: 2002 IV, page 51
10627 (* interval verification by Ferguson *)
10630 [((#8.0), x4, (square (#3.2)));
10631 ((#8.0), x4', (square (#3.2)))
10634 (vor_0_x (#4.0) (#4.0) (#4.0) x4 (#4.0) (#4.0)) +.
10635 (vor_0_x (#4.0) (#4.0) (#4.0) x4' (#4.0) (#4.0)) +.
10636 (vor_0_x (#4.0) (#4.0) (#4.0) x4 x4' (#4.0))) <.
10637 ( (--. (#0.05704)) +. (--. (#0.008))))`;;
10641 (* interval verification by Ferguson *)
10644 [((#8.0), x4, (square (#3.2)));
10645 ((#8.0), x4', (square (#3.2)))
10647 ( ( (tau_0_x (#4.0) (#4.0) (#4.0) x4 (#4.0) (#4.0)) +.
10648 (tau_0_x (#4.0) (#4.0) (#4.0) x4' (#4.0) (#4.0)) +.
10649 (tau_0_x (#4.0) (#4.0) (#4.0) x4 x4' (#4.0))) >.
10650 ( (#0.27113) +. (#0.008)))`;;
10654 (* interval verification by Ferguson *)
10657 [((#8.0), x4, (square (#3.2)));
10658 ((#8.0), x5, (square (#3.2)));
10659 ((#8.0), x6, (square (#3.2)))
10661 ( (vor_0_x (#4.0) (#4.0) (#4.0) x4 x5 x6) <.
10662 ( ( (--. (#2.0)) *. (#0.008)) +. (--. (#0.11408)) +.
10663 ( (--. (#3.0)) *. (#0.0461))))`;;
10667 (* interval verification by Ferguson *)
10670 [((#8.0), x4, (square (#3.2)));
10671 ((#8.0), x5, (square (#3.2)));
10672 ((#8.0), x6, (square (#3.2)))
10674 ( (tau_0_x (#4.0) (#4.0) (#4.0) x4 x5 x6) >.
10675 ( (#0.41056) +. (#0.06688)))`;;
10677 let CIVA21_692155251 = list_mk_conj
10678 [ I_275286804;I_627654828;I_995177961;I_735892048;];;
10681 LOC: 2002 IV, page 51
10685 In $\A_{22}$ and $\A_{23}$, $y_1\in [2t_0,2\sqrt2]$,
10686 $y_4\in[2\sqrt2,3.2]$, and $\dih<2.46$. $\vor_0(Q)$ denotes the
10687 truncated Voronoi function on the union of an anchored simplex and an
10688 adjacent special simplex. Let $S'$ be the special simplex. By
10689 deformations, $y_1(S')\in\{2,2t_0\}$. If $y_1(S')=2t_0$, the
10690 verifications follow from $\A_6$ and $\vor_0(S')\le0$. We may assume
10691 that $y_1(S')=2$. Also by deformations, $y_5(S')=y_6(S')=2$.
10696 (* ineq changed from weak to strick on dih *)
10697 (* interval verification by Ferguson *)
10700 [(square_2t0,x1,(#8.0));
10701 ((#4.0), x2, square_2t0);
10702 ((#4.0), x3, square_2t0);
10703 ((#8.0), x4, (square (#3.2)));
10704 ((#4.0), x5, square_2t0);
10705 ((#4.0), x6, square_2t0)
10707 (((vor_0_x x1 x2 x3 x4 x5 x6) + (vor_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
10708 <. (--(#3.58) + (#2.28501)*(dih_x x1 x2 x3 x4 x5 x6))) \/
10709 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
10711 (* ineq changed from weak to strick on dih *)
10712 (* interval verification by Ferguson *)
10715 [(square_2t0,x1,(#8.0));
10716 ((#4.0), x2, square_2t0);
10717 ((#4.0), x3, square_2t0);
10718 ((#8.0), x4, (square (#3.2)));
10719 ((#4.0), x5, square_2t0);
10720 ((#4.0), x6, square_2t0)
10722 (((vor_0_x x1 x2 x3 x4 x5 x6) + (vor_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
10723 <. (--(#2.715) + (#1.67382)*(dih_x x1 x2 x3 x4 x5 x6))) \/
10724 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
10726 (* ineq changed from weak to strick on dih *)
10727 (* interval verification by Ferguson *)
10730 [(square_2t0,x1,(#8.0));
10731 ((#4.0), x2, square_2t0);
10732 ((#4.0), x3, square_2t0);
10733 ((#8.0), x4, (square (#3.2)));
10734 ((#4.0), x5, square_2t0);
10735 ((#4.0), x6, square_2t0)
10737 (((vor_0_x x1 x2 x3 x4 x5 x6) + (vor_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
10738 <. (--(#1.517) + (#0.8285)*(dih_x x1 x2 x3 x4 x5 x6))) \/
10739 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
10741 (* ineq changed from weak to strick on dih *)
10742 (* interval verification by Ferguson *)
10745 [(square_2t0,x1,(#8.0));
10746 ((#4.0), x2, square_2t0);
10747 ((#4.0), x3, square_2t0);
10748 ((#8.0), x4, (square (#3.2)));
10749 ((#4.0), x5, square_2t0);
10750 ((#4.0), x6, square_2t0)
10752 (((vor_0_x x1 x2 x3 x4 x5 x6) + (vor_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
10753 <. (--(#0.858) + (#0.390925)*(dih_x x1 x2 x3 x4 x5 x6))) \/
10754 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
10756 (* ineq changed from weak to strick on dih *)
10757 (* interval verification by Ferguson *)
10760 [(square_2t0,x1,(#8.0));
10761 ((#4.0), x2, square_2t0);
10762 ((#4.0), x3, square_2t0);
10763 ((#8.0), x4, (square (#3.2)));
10764 ((#4.0), x5, square_2t0);
10765 ((#4.0), x6, square_2t0)
10767 (((vor_0_x x1 x2 x3 x4 x5 x6) + (vor_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
10768 <. (--(#0.358) + (#0.009)+ (#0.12012)*(dih_x x1 x2 x3 x4 x5 x6))) \/
10769 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
10771 (* ineq changed from weak to strick on dih *)
10772 (* interval verification by Ferguson *)
10775 [(square_2t0,x1,(#8.0));
10776 ((#4.0), x2, square_2t0);
10777 ((#4.0), x3, square_2t0);
10778 ((#8.0), x4, (square (#3.2)));
10779 ((#4.0), x5, square_2t0);
10780 ((#4.0), x6, square_2t0)
10782 (((vor_0_x x1 x2 x3 x4 x5 x6) + (vor_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
10783 <. (--(#0.186) + (#0.009)+ (#0.0501)*(dih_x x1 x2 x3 x4 x5 x6))) \/
10784 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
10789 (* interval verification by Ferguson *)
10792 [(square_2t0, x1, (#8.0));
10793 ((#4.0), x2, square_2t0);
10794 ((#4.0), x3, square_2t0);
10796 ((#4.0), x4, square_2t0);
10797 (square_2t0, x5, square_2t0);
10798 ((#4.0), x6, square_2t0)
10800 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. ( ( (--. (#3.58)) / (#2.0)) +. ( (#2.28501) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
10804 (* interval verification by Ferguson *)
10807 [(square_2t0, x1, (#8.0));
10808 ((#4.0), x2, square_2t0);
10809 ((#4.0), x3, square_2t0);
10811 ((#4.0), x4, square_2t0);
10812 (square_2t0, x5, square_2t0);
10813 ((#4.0), x6, square_2t0)
10815 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. ( ( (--. (#2.715)) / (#2.0)) +. ( (#1.67382) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
10819 (* interval verification by Ferguson *)
10822 [(square_2t0, x1, (#8.0));
10823 ((#4.0), x2, square_2t0);
10824 ((#4.0), x3, square_2t0);
10826 ((#4.0), x4, square_2t0);
10827 (square_2t0, x5, square_2t0);
10828 ((#4.0), x6, square_2t0)
10830 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. ( ( (--. (#1.517)) / (#2.0)) +. ( (#0.8285) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
10834 (* interval verification by Ferguson *)
10837 [(square_2t0, x1, (#8.0));
10838 ((#4.0), x2, square_2t0);
10839 ((#4.0), x3, square_2t0);
10841 ((#4.0), x4, square_2t0);
10842 (square_2t0, x5, square_2t0);
10843 ((#4.0), x6, square_2t0)
10845 ( (vor_0_x x1 x2 x3 x4 x5 x6) <. ( ( (--. (#0.858)) / (#2.0)) +. ( (#0.390925) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
10850 (* interval verification by Ferguson *)
10853 [(square_2t0, x1, (#8.0));
10854 ((#4.0), x2, square_2t0);
10855 ((#4.0), x3, square_2t0);
10857 ((#4.0), x4, square_2t0);
10858 (square_2t0, x5, square_2t0);
10859 ((#4.0), x6, square_2t0)
10861 ( (vor_0_x x1 x2 x3 x4 x5 x6) <.
10862 ( ( ( (--. (#0.358)) +. (#0.009)) / (#2.0)) +. ( (#0.12012) *. (dih_x x1 x2 x3 x4 x5 x6)) +.
10863 ( (#0.2) *. ( (dih_x x1 x2 x3 x4 x5 x6) +. (--. (#1.23))))))`;;
10868 (* interval verification by Ferguson *)
10871 [(square_2t0, x1, (#8.0));
10872 ((#4.0), x2, square_2t0);
10873 ((#4.0), x3, square_2t0);
10875 ((#4.0), x4, square_2t0);
10876 (square_2t0, x5, square_2t0);
10877 ((#4.0), x6, square_2t0)
10879 ( (vor_0_x x1 x2 x3 x4 x5 x6) <.
10880 ( ( ( (--. (#0.186)) +. (#0.009)) / (#2.0)) +. ( (#0.0501) *. (dih_x x1 x2 x3 x4 x5 x6)) +.
10881 ( (#0.2) *. ( (dih_x x1 x2 x3 x4 x5 x6) +. (--. (#1.23))))))`;;
10883 let CIVA22_485049042 = list_mk_conj
10884 [ I_53502142;I_134398524;I_371491817;I_832922998;
10885 I_724796759;I_431940343;I_980721294;I_989564937;
10886 I_263355808;I_445132132;I_806767374;I_511038592;];;
10890 LOC: 2002 IV, page 51--52
10893 Note from text (appearing after the first seven) :
10895 Let $S'$ be the special simplex. By deformations, we have
10896 $y_5(S')=y_6(S')=2$, and $y_1(S')\in\{2,2t_0\}$. If $y_1(S')=2t_0$, and
10897 $y_4(S')\le3$, the inequalities listed above follow from Section~$\A_7$
10898 and the inequality #8 \refno{66753311}
10900 Similarly, the result follows if $y_2$ or $y_3\ge2.2$ from the
10901 inequality #9 \refno{762922223}
10904 Because of these reductions, we may assume in the first batch of
10905 inequalities of $\A_{23}$ that when $y_1(S')\ne2$, we have that
10906 $y_1(S')=2t_0$, $y_5(S')=y_6(S')=2$, $y_4\in[3,3.2]$,
10907 $y_2(S'),y_3(S')\le2.2$. In all but {\tt (371464244)} and {\tt
10908 (657011065)}, if $y_1(S')=2t_0$, we prove the inequality with
10909 $\tau_0(S')$ replaced with its lower bound $0$.
10911 Again if the cross-diagonal is $2t_0$, we break $Q$ in the other
10912 direction. Let $S''$ be an upright quarter with $y_5=2t_0$. Set $\tau_0
10913 = \tau_0(S'')$. We have ...
10921 (* interval verification by Ferguson *)
10924 [(square_2t0,x1,(#8.0));
10925 ((#4.0), x2, square_2t0);
10926 ((#4.0), x3, square_2t0);
10927 ((#8.0), x4,(square (#3.2)));
10928 ((#4.0), x5, square_2t0);
10929 ((#4.0), x6, square_2t0)
10932 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
10933 (tau_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
10935 (--(#3.48) + (#2.1747)*(dih_x x1 x2 x3 x4 x5 x6))) \/
10936 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
10938 (* interval verification by Ferguson *)
10941 [(square_2t0,x1,(#8.0));
10942 ((#4.0), x2, square_2t0);
10943 ((#4.0), x3, square_2t0);
10944 ((#8.0), x4,(square (#3.2)));
10945 ((#4.0), x5, square_2t0);
10946 ((#4.0), x6, square_2t0)
10949 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
10950 (tau_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
10952 (--(#3.06) + (#1.87427)*(dih_x x1 x2 x3 x4 x5 x6))) \/
10953 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
10955 (* interval verification by Ferguson *)
10958 [(square_2t0,x1,(#8.0));
10959 ((#4.0), x2, square_2t0);
10960 ((#4.0), x3, square_2t0);
10961 ((#8.0), x4,(square (#3.2)));
10962 ((#4.0), x5, square_2t0);
10963 ((#4.0), x6, square_2t0)
10966 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
10967 (tau_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
10969 (--(#1.58) + (#0.83046)*(dih_x x1 x2 x3 x4 x5 x6))) \/
10970 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
10972 (* interval verification by Ferguson *)
10975 [(square_2t0,x1,(#8.0));
10976 ((#4.0), x2, square_2t0);
10977 ((#4.0), x3, square_2t0);
10978 ((#8.0), x4,(square (#3.2)));
10979 ((#4.0), x5, square_2t0);
10980 ((#4.0), x6, square_2t0)
10983 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
10984 (tau_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
10986 (--(#1.06) + (#0.48263)*(dih_x x1 x2 x3 x4 x5 x6))) \/
10987 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
10989 (* interval verification by Ferguson *)
10992 [(square_2t0,x1,(#8.0));
10993 ((#4.0), x2, square_2t0);
10994 ((#4.0), x3, square_2t0);
10995 ((#8.0), x4,(square (#3.2)));
10996 ((#4.0), x5, square_2t0);
10997 ((#4.0), x6, square_2t0)
11000 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
11001 (tau_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
11003 (--(#0.83) + (#0.34833)*(dih_x x1 x2 x3 x4 x5 x6))) \/
11004 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
11006 (* interval verification by Ferguson *)
11009 [(square_2t0,x1,(#8.0));
11010 ((#4.0), x2, square_2t0);
11011 ((#4.0), x3, square_2t0);
11012 ((#8.0), x4,(square (#3.2)));
11013 ((#4.0), x5, square_2t0);
11014 ((#4.0), x6, square_2t0)
11017 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
11018 (tau_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
11020 (--(#0.5) + (#0.1694)*(dih_x x1 x2 x3 x4 x5 x6))) \/
11021 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
11023 (* interval verification by Ferguson *)
11026 [(square_2t0,x1,(#8.0));
11027 ((#4.0), x2, square_2t0);
11028 ((#4.0), x3, square_2t0);
11029 ((#8.0), x4,(square (#3.2)));
11030 ((#4.0), x5, square_2t0);
11031 ((#4.0), x6, square_2t0)
11034 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
11035 (tau_0_x (#4.0) x2 x3 x4 (#4.0) (#4.0))
11037 (--(#0.29) + (#0.0014)+ (#0.0822)*(dih_x x1 x2 x3 x4 x5 x6))) \/
11038 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
11041 (* interval verification by Ferguson *)
11044 [(square_2t0,x1,(#8.0));
11045 ((#4.0), x2, (square (#2.2)));
11046 ((#4.0), x3, (square (#2.2)));
11047 ((square (#3.0)), x4,(square (#3.2)));
11048 ((#4.0), x5, square_2t0);
11049 ((#4.0), x6, square_2t0)
11052 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
11053 (tau_0_x (square_2t0) x2 x3 x4 (#4.0) (#4.0))
11055 (--(#3.48) + (#2.1747)*(dih_x x1 x2 x3 x4 x5 x6))) \/
11056 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
11058 (* interval verification by Ferguson *)
11061 [(square_2t0,x1,(#8.0));
11062 ((#4.0), x2, (square (#2.2)));
11063 ((#4.0), x3, (square (#2.2)));
11064 ((square (#3.0)), x4,(square (#3.2)));
11065 ((#4.0), x5, square_2t0);
11066 ((#4.0), x6, square_2t0)
11069 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
11070 (tau_0_x (square_2t0) x2 x3 x4 (#4.0) (#4.0))
11072 (--(#3.06) + (#1.87427)*(dih_x x1 x2 x3 x4 x5 x6))) \/
11073 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
11075 (* interval verification by Ferguson *)
11078 [(square_2t0,x1,(#8.0));
11079 ((#4.0), x2, (square (#2.2)));
11080 ((#4.0), x3, (square (#2.2)));
11081 ((square (#3.0)), x4,(square (#3.2)));
11082 ((#4.0), x5, square_2t0);
11083 ((#4.0), x6, square_2t0)
11086 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
11087 (tau_0_x (square_2t0) x2 x3 x4 (#4.0) (#4.0))
11089 (--(#1.58) + (#0.83046)*(dih_x x1 x2 x3 x4 x5 x6))) \/
11090 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
11092 (* interval verification by Ferguson *)
11095 [(square_2t0,x1,(#8.0));
11096 ((#4.0), x2, (square (#2.2)));
11097 ((#4.0), x3, (square (#2.2)));
11098 ((square (#3.0)), x4,(square (#3.2)));
11099 ((#4.0), x5, square_2t0);
11100 ((#4.0), x6, square_2t0)
11103 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
11104 (tau_0_x (square_2t0) x2 x3 x4 (#4.0) (#4.0))
11106 (--(#1.06) + (#0.48263)*(dih_x x1 x2 x3 x4 x5 x6))) \/
11107 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
11109 (* interval verification by Ferguson *)
11112 [(square_2t0,x1,(#8.0));
11113 ((#4.0), x2, (square (#2.2)));
11114 ((#4.0), x3, (square (#2.2)));
11115 ((square (#3.0)), x4,(square (#3.2)));
11116 ((#4.0), x5, square_2t0);
11117 ((#4.0), x6, square_2t0)
11120 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
11121 (tau_0_x (square_2t0) x2 x3 x4 (#4.0) (#4.0))
11123 (--(#0.83) + (#0.34833)*(dih_x x1 x2 x3 x4 x5 x6))) \/
11124 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
11126 (* interval verification by Ferguson *)
11132 [(square_2t0,x1,(#8.0));
11133 ((#4.0), x2, (square (#2.2)));
11134 ((#4.0), x3, (square (#2.2)));
11135 ((square (#3.0)), x4,(square (#3.2)));
11136 ((#4.0), x5, square_2t0);
11137 ((#4.0), x6, square_2t0)
11140 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
11141 (tau_0_x (square_2t0) x2 x3 x4 (#4.0) (#4.0))
11143 (--(#0.5) + (#0.1694)*(dih_x x1 x2 x3 x4 x5 x6))) \/
11144 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
11146 (* interval verification by Ferguson *)
11147 let I_657011065_2 =
11149 [(square_2t0,x1,(#8.0));
11150 ((#4.0), x2, (square (#2.2)));
11151 ((#4.0), x3, (square (#2.2)));
11152 ((square (#3.0)), x4,(square (#3.2)));
11153 ((#4.0), x5, square_2t0);
11154 ((#4.0), x6, square_2t0)
11157 ((--(tau_0_x x1 x2 x3 x4 x5 x6) ) -
11158 (tau_0_x (square_2t0) x2 x3 x4 (#4.0) (#4.0))
11160 (--(#0.29) + (#0.0014)+ (#0.0822)*(dih_x x1 x2 x3 x4 x5 x6))) \/
11161 (dih_x x1 x2 x3 x4 x5 x6 >. (#2.46)))`;;
11165 (* interval verification by Ferguson *)
11166 (* id number corrected from 55753311 *)
11170 [ ((#4.0), x2, square_2t0);
11171 ((#4.0), x3, square_2t0);
11172 ((#8.0), x4,(square (#3.0)))
11175 (tau_0_x (square_2t0) x2 x3 x4 (#4.0) (#4.0) ) >. (#0.06585)
11178 (* interval verification by Ferguson *)
11180 CCC fixed domain 3/10/2008.
11181 Bound: 0.0658173454705
11183 Point: [4.09979901231, 4.0015878624, 9.8006368154]
11188 [ ((square (#2.2)), x2,square_2t0);
11189 ((#4.0), x3, square_2t0);
11190 ((square (#3.0)), x4,(square (#3.2)))
11193 (tau_0_x (square_2t0) x2 x3 x4 (#4.0) (#4.0) ) >. (#0.06585)
11197 (* calcs 10 -- 16 *)
11198 (* interval verification by Ferguson *)
11201 [(square_2t0, x1, (#8.0));
11202 ((#4.0), x2, square_2t0);
11203 ((#4.0), x3, square_2t0);
11205 ((#4.0), x4, square_2t0);
11206 (square_2t0, x5, square_2t0);
11207 ((#4.0), x6, square_2t0)
11209 ( ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) +.
11210 ( (#0.06585) / (#2.0))) <.
11211 ( ( (--. (#3.48)) / (#2.0)) +.
11212 ( (#2.1747) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
11214 (* interval verification by Ferguson *)
11217 [(square_2t0, x1, (#8.0));
11218 ((#4.0), x2, square_2t0);
11219 ((#4.0), x3, square_2t0);
11221 ((#4.0), x4, square_2t0);
11222 (square_2t0, x5, square_2t0);
11223 ((#4.0), x6, square_2t0)
11225 ( ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) +.
11226 ( (#0.06585) / (#2.0))) <.
11227 ( ( (--. (#3.06)) / (#2.0)) +.
11228 ( (#1.87427) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
11230 (* interval verification by Ferguson *)
11233 [(square_2t0, x1, (#8.0));
11234 ((#4.0), x2, square_2t0);
11235 ((#4.0), x3, square_2t0);
11237 ((#4.0), x4, square_2t0);
11238 (square_2t0, x5, square_2t0);
11239 ((#4.0), x6, square_2t0)
11241 ( ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) +.
11242 ( (#0.06585) / (#2.0))) <.
11243 ( ( (--. (#1.58)) / (#2.0)) +.
11244 ( (#0.83046) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
11246 (* interval verification by Ferguson *)
11249 [(square_2t0, x1, (#8.0));
11250 ((#4.0), x2, square_2t0);
11251 ((#4.0), x3, square_2t0);
11253 ((#4.0), x4, square_2t0);
11254 (square_2t0, x5, square_2t0);
11255 ((#4.0), x6, square_2t0)
11257 ( ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) +.
11258 ( (#0.06585) / (#2.0))) <.
11259 ( ( (--. (#1.06)) / (#2.0)) +.
11260 ( (#0.48263) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
11264 (* interval verification by Ferguson *)
11267 [(square_2t0, x1, (#8.0));
11268 ((#4.0), x2, square_2t0);
11269 ((#4.0), x3, square_2t0);
11271 ((#4.0), x4, square_2t0);
11272 (square_2t0, x5, square_2t0);
11273 ((#4.0), x6, square_2t0)
11275 ( ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) +.
11276 ( (#0.06585) / (#2.0))) <.
11277 ( ( (--. (#0.83)) / (#2.0)) +.
11278 ( (#0.34833) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
11282 (* interval verification by Ferguson *)
11285 [(square_2t0, x1, (#8.0));
11286 ((#4.0), x2, square_2t0);
11287 ((#4.0), x3, square_2t0);
11289 ((#4.0), x4, square_2t0);
11290 (square_2t0, x5, square_2t0);
11291 ((#4.0), x6, square_2t0)
11293 ( ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) +.
11294 ( (#0.06585) / (#2.0))) <.
11295 ( ( (--. (#0.50)) / (#2.0)) +.
11296 ( (#0.1694) *. (dih_x x1 x2 x3 x4 x5 x6)) +.
11297 ( (#0.03) *. ( (dih_x x1 x2 x3 x4 x5 x6) +.
11298 (--. (#1.23))))))`;;
11302 (* interval verification by Ferguson *)
11305 [(square_2t0, x1, (#8.0));
11306 ((#4.0), x2, square_2t0);
11307 ((#4.0), x3, square_2t0);
11309 ((#4.0), x4, square_2t0);
11310 (square_2t0, x5, square_2t0);
11311 ((#4.0), x6, square_2t0)
11313 ( ( (( --. ) (tau_0_x x1 x2 x3 x4 x5 x6)) +.
11314 ( (#0.06585) / (#2.0))) <.
11315 ( ( (--. (#0.29)) / (#2.0)) +.
11316 ( (#0.0014) / (#2.0)) +.
11317 ( (#0.0822) *. (dih_x x1 x2 x3 x4 x5 x6)) +.
11318 ( (#0.2) *. ( (dih_x x1 x2 x3 x4 x5 x6) +.
11319 (--. (#1.23))))))`;;
11321 let CIVA23_209361863= list_mk_conj
11322 [ I_4591018_1;I_193728878_1;I_2724096_1;I_213514168_1;
11323 I_750768322_1;I_371464244_1;I_657011065_1;I_4591018_2;
11324 I_193728878_2;I_2724096_2;I_213514168_2;I_750768322_2;
11325 I_371464244_2;I_657011065_2 ;I_66753311;I_762922223;
11326 I_953023504;I_887276655;I_246315515;I_784421604;
11327 I_258632246;I_404164527;I_163088471;];;
11330 LOC: 2002 IV, page 52
11335 (* interval verification in partK.cc *)
11336 (* interval verification by Ferguson *)
11339 [(square_2t0, x1, (#8.0));
11340 ((#4.0), x2, (#4.0));
11341 ((#4.0), x3, square_2t0);
11343 ((#4.0), x4, (#4.0));
11344 ((#4.0), x5, square_2t0);
11345 (square_2t0, x6, (square (#2.75)))
11347 ( ( (tau_0_x x1 x2 x3 x4 x5 x6) +.
11348 ( (#0.0822) *. (dih_x x1 x2 x3 x4 x5 x6))) >. (#0.159))`;;
11352 (* interval verification in partK.cc *)
11353 (* interval verification by Ferguson *)
11357 [(square_2t0, x1, (#8.0));
11358 (square_2t0, x2, square_2t0);
11359 ((#4.0), x3, square_2t0);
11361 ((#4.0), x4, (#4.0));
11362 ((#4.0), x5, square_2t0);
11363 (square_2t0, x6, square_4t0)
11365 (( dih_x x1 x2 x3 x4 x5 x6 <. (#1.23)) \/
11366 (delta_x x1 x2 x3 x4 x5 x6 <. (#0.0)))`;;
11370 (* interval verification in partK.cc *)
11371 (* interval verification by Ferguson *)
11375 [(square_2t0, x1, (#8.0));
11376 ((#4.0), x2, (#4.0));
11377 ((#4.0), x3, square_2t0);
11379 ((#4.0), x4, (#4.0));
11380 ((#4.0), x5, square_2t0);
11381 ((square (#2.75)), x6, square_4t0)
11383 (( (dih_x x1 x2 x3 x4 x5 x6) <. (#1.23)) \/
11384 (delta_x x1 x2 x3 x4 x5 x6 <. (#0.0)))`;;
11386 let CIVA24_835344007= list_mk_conj
11387 [ I_968721007;I_783968228;I_745174731;];;
11392 LOC: 2002 III, page 14.
11398 (* moved 586468779 to inequality_spec.ml *)
11403 (* moved 984463800 to inequality_spec.ml *)
11408 (* moved 208809199 to inequality_spec.ml *)
11415 [((#4.0), x1, square_2t0);
11416 ((#4.0), x2, square_2t0);
11417 ((#4.0), x3, square_2t0);
11418 ((#4.0), x4, square_2t0);
11419 ((#4.0), x5, square_2t0);
11420 ((#4.0), x6, square_2t0)
11422 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
11423 ( ( (--. (#0.37642101)) *. (sol_x x1 x2 x3 x4 x5 x6)) +. (#0.287389)))`;;
11429 [((#4.0), x1, square_2t0);
11430 ((#4.0), x2, square_2t0);
11431 ((#4.0), x3, square_2t0);
11433 ((#4.0), x4, square_2t0);
11434 ((#4.0), x5, square_2t0);
11435 ((#4.0), x6, square_2t0)
11437 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
11438 ( ( (#0.446634) *. (sol_x x1 x2 x3 x4 x5 x6)) +. (--. (#0.190249))))`;;
11444 [((#4.0), x1, square_2t0);
11445 ((#4.0), x2, square_2t0);
11446 ((#4.0), x3, square_2t0);
11447 ((#4.0), x4, square_2t0);
11448 ((#4.0), x5, square_2t0);
11449 ((#4.0), x6, square_2t0)
11451 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
11452 ( ( (--. (#0.419351)) *. (sol_x x1 x2 x3 x4 x5 x6)) +. (#0.2856354) +. (#0.001)))`;;
11457 SKIP equation 7. (sigma(quad) <= 0)
11458 This is proved as a theorem and is not really an
11459 interval arithmetic result.
11464 LOC: 2002 III, page 14.
11471 [((#4.0), x1, square_2t0);
11472 ((#4.0), x2, square_2t0);
11473 ((#4.0), x3, square_2t0);
11475 ((#4.0), x4, square_2t0);
11476 ((#4.0), x5, square_2t0);
11477 ((#4.0), x6, square_2t0)
11479 ( (sol_x x1 x2 x3 x4 x5 x6) >.
11480 ( (#0.551285) +. ( (#0.199235) *. ( (sqrt x4) +. (sqrt x5) +. (sqrt x6) +. (--. (#6.0)))) +.
11481 ( (--. (#0.377076)) *. ( (sqrt x1) +. (sqrt x2) +. (sqrt x3) +. (--. (#6.0))))))`;;
11487 [((#4.0), x1, square_2t0);
11488 ((#4.0), x2, square_2t0);
11489 ((#4.0), x3, square_2t0);
11491 ((#4.0), x4, square_2t0);
11492 ((#4.0), x5, square_2t0);
11493 ((#4.0), x6, square_2t0)
11495 ( (sol_x x1 x2 x3 x4 x5 x6) <.
11496 ( (#0.551286) +. ( (#0.320937) *. ( (sqrt x4) +. (sqrt x5) +. (sqrt x6) +. (--. (#6.0)))) +.
11497 ( (--. (#0.152679)) *. ( (sqrt x1) +. (sqrt x2) +. (sqrt x3) +. (--. (#6.0))))))`;;
11503 [((#4.0), x1, square_2t0);
11504 ((#4.0), x2, square_2t0);
11505 ((#4.0), x3, square_2t0);
11507 ((#4.0), x4, square_2t0);
11508 ((#4.0), x5, square_2t0);
11509 ((#4.0), x6, square_2t0)
11511 ( (dih_x x1 x2 x3 x4 x5 x6) >.
11512 ( (#1.23095) +. ( (--. (#0.359894)) *. ( (sqrt x2) +. (sqrt x3) +. (sqrt x5) +. (sqrt x6) +. (--. (#8.0)))) +.
11513 ( (#0.003) *. ( (sqrt x1) +. (--. (#2.0)))) +. ( (#0.685) *. ( (sqrt x4) +. (--. (#2.0))))))`;;
11520 [((#4.0), x1, square_2t0);
11521 ((#4.0), x2, square_2t0);
11522 ((#4.0), x3, square_2t0);
11524 ((#4.0), x4, square_2t0);
11525 ((#4.0), x5, square_2t0);
11526 ((#4.0), x6, square_2t0)
11528 ( (dih_x x1 x2 x3 x4 x5 x6) <.
11529 ( (#1.23096) +. ( (--. (#0.153598)) *. ( (sqrt x2) +. (sqrt x3) +. (sqrt x5) +. (sqrt x6) +. (--. (#8.0)))) +.
11530 ( (#0.498) *. ( (sqrt x1) +. (--. (#2.0)))) +. ( (#0.76446) *. ( (sqrt x4) +. (--. (#2.0))))))`;;
11536 [((#4.0), x1, square_2t0);
11537 ((#4.0), x2, square_2t0);
11538 ((#4.0), x3, square_2t0);
11539 ((#4.0), x4, square_2t0);
11540 ((#4.0), x5, square_2t0);
11541 ((#4.0), x6, square_2t0)
11543 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
11545 ( (--. (#0.10857)) *. ( (sqrt x1) +. (sqrt x2) +. (sqrt x3) +. (sqrt x4) +. (sqrt x5) +. (sqrt x6) +. (--. (#12.0))))))`;;
11551 [((#4.0), x1, square_2t0);
11552 ((#4.0), x2, square_2t0);
11553 ((#4.0), x3, square_2t0);
11554 ((#4.0), x4, square_2t0);
11555 ((#4.0), x5, square_2t0);
11556 ((#4.0), x6, square_2t0)
11558 ( ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) +. ( (#0.419351) *. (sol_x x1 x2 x3 x4 x5 x6))) <.
11559 ( (#0.28665) +. ( (--. (#0.2)) *. ( (sqrt x1) +. (sqrt x2) +. (sqrt x3) +. (--. (#6.0))))))`;;
11565 [((#4.0), x1, square_2t0);
11566 ((#4.0), x2, square_2t0);
11567 ((#4.0), x3, square_2t0);
11568 ((#4.0), x4, square_2t0);
11569 ((#4.0), x5, square_2t0);
11570 ((#4.0), x6, square_2t0)
11572 ( (sigma1_qrtet_x x1 x2 x3 x4 x5 x6) <.
11573 ( (#0.000001) +. ( (--. (#0.129119)) *. ( (sqrt x4) +. (sqrt x5) +. (sqrt x6) +. (--. (#6.0)))) +.
11574 ( (--. (#0.0845696)) *. ( (sqrt x1) +. (sqrt x2) +. (sqrt x3) +. (--. (#6.0))))))`;;
11579 LOC: 2002 III, page 14--15
11583 (* interval verification in part3.cc, but labeled there as C619245724 *)
11586 [((#4.0), x1, square_2t0);
11587 ((#4.0), x2, square_2t0);
11588 ((#4.0), x3, square_2t0);
11589 ((#4.0), x4, square_2t0);
11590 ((#4.0), x5, square_2t0);
11591 ((#4.0), x6, square_2t0)
11593 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
11594 ( ( (#0.37898) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (--. (#0.4111))))`;;
11597 (* interval verification in part3.cc, but labeled there as C678284947 *)
11600 [((#4.0), x1, square_2t0);
11601 ((#4.0), x2, square_2t0);
11602 ((#4.0), x3, square_2t0);
11603 ((#4.0), x4, square_2t0);
11604 ((#4.0), x5, square_2t0);
11605 ((#4.0), x6, square_2t0)
11607 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
11608 ( ( (--. (#0.142)) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (#0.23021)))`;;
11611 (* interval verification in part3.cc, but labeled there as C970731712 *)
11614 [((#4.0), x1, square_2t0);
11615 ((#4.0), x2, square_2t0);
11616 ((#4.0), x3, square_2t0);
11617 ((#4.0), x4, square_2t0);
11618 ((#4.0), x5, square_2t0);
11619 ((#4.0), x6, square_2t0)
11621 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
11622 ( ( (--. (#0.3302)) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (#0.5353)))`;;
11625 (* interval verification in part3.cc, but labeled there as C921602098 *)
11628 [((#4.0), x1, square_2t0);
11629 ((#4.0), x2, square_2t0);
11630 ((#4.0), x3, square_2t0);
11631 ((#4.0), x4, square_2t0);
11632 ((#4.0), x5, square_2t0);
11633 ((#4.0), x6, square_2t0)
11635 ( (sigma1_qrtet_x x1 x2 x3 x4 x5 x6) <.
11636 ( ( (#0.3897) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (--. (#0.4666))))`;;
11639 (* interval verification in part3.cc, but labeled there as C338482233 *)
11642 [((#4.0), x1, square_2t0);
11643 ((#4.0), x2, square_2t0);
11644 ((#4.0), x3, square_2t0);
11646 ((#4.0), x4, square_2t0);
11647 ((#4.0), x5, square_2t0);
11648 ((#4.0), x6, square_2t0)
11650 ( (sigma1_qrtet_x x1 x2 x3 x4 x5 x6) <.
11651 ( ( (#0.2993) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (--. (#0.3683))))`;;
11654 (* interval verification in part3.cc, but labeled there as C47923787 *)
11655 (* moved 53415898 to inequality_spec.ml *)
11659 (* interval verification in part3.cc, but labeled there as C156673846 *)
11662 [((#4.0), x1, square_2t0);
11663 ((#4.0), x2, square_2t0);
11664 ((#4.0), x3, square_2t0);
11666 ((#4.0), x4, square_2t0);
11667 ((#4.0), x5, square_2t0);
11668 ((#4.0), x6, square_2t0)
11670 ( (sigma1_qrtet_x x1 x2 x3 x4 x5 x6) <.
11671 ( ( (--. (#0.1689)) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (#0.208)))`;;
11674 (* interval verification in part3.cc, but labeled there as C363044842 *)
11677 [((#4.0), x1, square_2t0);
11678 ((#4.0), x2, square_2t0);
11679 ((#4.0), x3, square_2t0);
11681 ((#4.0), x4, square_2t0);
11682 ((#4.0), x5, square_2t0);
11683 ((#4.0), x6, square_2t0)
11685 ( (sigma1_qrtet_x x1 x2 x3 x4 x5 x6) <.
11686 ( ( (--. (#0.2529)) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (#0.3442)))`;;
11689 (* interval verification in part3.cc, but labeled there as C68229886 *)
11692 [((#4.0), x1, square_2t0);
11693 ((#4.0), x2, square_2t0);
11694 ((#4.0), x3, square_2t0);
11696 ((#4.0), x4, square_2t0);
11697 ((#4.0), x5, square_2t0);
11698 ((#4.0), x6, square_2t0)
11700 ( (sigma32_qrtet_x x1 x2 x3 x4 x5 x6) <.
11701 ( ( (#0.4233) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (--. (#0.5974))))`;;
11704 (* interval verification in part3.cc, but labeled there as C996335124 *)
11707 [((#4.0), x1, square_2t0);
11708 ((#4.0), x2, square_2t0);
11709 ((#4.0), x3, square_2t0);
11711 ((#4.0), x4, square_2t0);
11712 ((#4.0), x5, square_2t0);
11713 ((#4.0), x6, square_2t0)
11715 ( (sigma32_qrtet_x x1 x2 x3 x4 x5 x6) <.
11716 ( ( (#0.1083) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (--. (#0.255))))`;;
11719 (* interval verification in part3.cc, but labeled there as C722658871 *)
11722 [((#4.0), x1, square_2t0);
11723 ((#4.0), x2, square_2t0);
11724 ((#4.0), x3, square_2t0);
11726 ((#4.0), x4, square_2t0);
11727 ((#4.0), x5, square_2t0);
11728 ((#4.0), x6, square_2t0)
11730 ( (sigma32_qrtet_x x1 x2 x3 x4 x5 x6) <.
11731 ( ( (--. (#0.0953)) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (--. (#0.0045))))`;;
11734 (* interval verification in part3.cc, but labeled there as C226224557 *)
11737 [((#4.0), x1, square_2t0);
11738 ((#4.0), x2, square_2t0);
11739 ((#4.0), x3, square_2t0);
11741 ((#4.0), x4, square_2t0);
11742 ((#4.0), x5, square_2t0);
11743 ((#4.0), x6, square_2t0)
11745 ( (sigma32_qrtet_x x1 x2 x3 x4 x5 x6) <.
11746 ( ( (--. (#0.1966)) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (#0.1369)))`;;
11749 (* interval verification in part3.cc, but labeled there as C914585134 *)
11752 [((#4.0), x1, square_2t0);
11753 ((#4.0), x2, square_2t0);
11754 ((#4.0), x3, square_2t0);
11755 ((#4.0), x4, square_2t0);
11756 ((#4.0), x5, square_2t0);
11757 ((#4.0), x6, square_2t0)
11759 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
11760 ( ( (--. (#0.419351)) *. (sol_x x1 x2 x3 x4 x5 x6)) +.
11761 ( (#0.796456) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (--. (#0.5786316))))`;;
11764 (* interval verification in part3.cc, but labeled there as C296182719 *)
11767 [((#4.0), x1, square_2t0);
11768 ((#4.0), x2, square_2t0);
11769 ((#4.0), x3, square_2t0);
11771 ((#4.0), x4, square_2t0);
11772 ((#4.0), x5, square_2t0);
11773 ((#4.0), x6, square_2t0)
11775 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
11776 ( ( (--. (#0.419351)) *. (sol_x x1 x2 x3 x4 x5 x6)) +.
11777 ( (#0.0610397) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (#0.211419)))`;;
11780 (* interval verification in part3.cc, but labeled there as C538860011 *)
11783 [((#4.0), x1, square_2t0);
11784 ((#4.0), x2, square_2t0);
11785 ((#4.0), x3, square_2t0);
11787 ((#4.0), x4, square_2t0);
11788 ((#4.0), x5, square_2t0);
11789 ((#4.0), x6, square_2t0)
11791 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
11792 ( ( (--. (#0.419351)) *. (sol_x x1 x2 x3 x4 x5 x6)) +.
11793 ( (--. (#0.0162028)) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (#0.308526)))`;;
11796 (* interval verification in part3.cc, but labeled there as C886673381 *)
11799 [((#4.0), x1, square_2t0);
11800 ((#4.0), x2, square_2t0);
11801 ((#4.0), x3, square_2t0);
11803 ((#4.0), x4, square_2t0);
11804 ((#4.0), x5, square_2t0);
11805 ((#4.0), x6, square_2t0)
11807 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
11808 ( ( (--. (#0.419351)) *. (sol_x x1 x2 x3 x4 x5 x6)) +.
11809 ( (--. (#0.0499559)) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (#0.35641)))`;;
11812 (* interval verification in part3.cc, but labeled there as C681494013 *)
11815 [((#4.0), x1, square_2t0);
11816 ((#4.0), x2, square_2t0);
11817 ((#4.0), x3, square_2t0);
11819 ((#4.0), x4, square_2t0);
11820 ((#4.0), x5, square_2t0);
11821 ((#4.0), x6, square_2t0)
11823 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
11824 ( ( (--. (#0.419351)) *. (sol_x x1 x2 x3 x4 x5 x6)) +.
11825 ( (--. (#0.64713719)) *. (dih_x x1 x2 x3 x4 x5 x6)) +. (#1.3225)))`;;
11828 let C_830854305 = list_mk_conj[
11829 J_539256862;J_864218323;J_776305271;J_927432550;J_221945658;
11830 J_53415898;J_106537269;J_254627291;J_170403135;J_802409438;
11831 J_195296574;J_16189133;J_584511898;J_98170671;J_868828815;
11832 J_809197575;J_73203677;];;
11836 SKIP statement about Quad clusters at end of Group_3
11837 This is Prop 4.1 and Prop 4.2 -- a long list of quad ineqs.
11838 These inequalities are in the file kep_inequalities2.ml
11843 [((#4.0), x1, square_2t0);
11844 ((#4.0), x2, square_2t0);
11845 ((#4.0), x3, square_2t0);
11847 (square_2t0, x4, (#8.0));
11848 ((#4.0), x5, square_2t0);
11849 ((#4.0), x6, square_2t0)
11851 ( ( ( (--. (#0.398)) *. ( (sqrt x2) +. (sqrt x3) +. (sqrt x5) +. (sqrt x6))) +.
11852 ( (#0.3257) *. (sqrt x1)) +. (( --. ) (dih_x x1 x2 x3 x4 x5 x6))) <. (--. (#4.14938)))`;;
11857 LOC: 2002 III, page 15.
11860 equation 5 is Prop 4.3 and Lemma 5.3.
11861 Proposition 4.3 appears in kep_inequalities2.ml.
11862 Lemma 5.3 is derived from other inequalities (Group_5), so it needn't
11863 be listed separately here.
11868 LOC: 2002 III, page 15.
11871 These are identical to the inequalities of 2002-III-Appendix 1:
11872 A.2.1--11, A.3.1--11, A.4.1--4, A.6.1--9, A.6.1'--8', A.8.1--3.
11873 These are all listed below.
11878 LOC: 2002 III, page 15.
11885 [((#4.0), x1, square_2t0);
11886 ((#4.0), x2, square_2t0);
11887 ((#4.0), x3, square_2t0);
11889 ((square (#2.1773)), x4, square_2t0);
11890 ((#4.0), x5, square_2t0);
11891 ((#4.0), x6, square_2t0)
11893 ( (tau_sigma_x x1 x2 x3 x4 x5 x6) >. ( (#0.55) *. pt))`;;
11899 [((#4.0), x1, square_2t0);
11900 ((#4.0), x2, square_2t0);
11901 ((#4.0), x3, square_2t0);
11903 ((square (#2.1773)), x4, square_2t0);
11904 ((square (#2.1773)), x5, square_2t0);
11905 ((#4.0), x6, square_2t0)
11907 ( (tau_sigma_x x1 x2 x3 x4 x5 x6) >. ( (#2.0) *. (#0.55) *. pt))`;;
11913 [((#4.0), x1, square_2t0);
11914 ((#4.0), x2, square_2t0);
11915 ((#4.0), x3, square_2t0);
11917 ((#4.0), x4, (square (#2.1773)));
11918 ((#4.0), x5, square_2t0);
11919 ((#4.0), x6, square_2t0)
11921 ( (tau_sigma_x x1 x2 x3 x4 x5 x6) >. ( (--. (#0.29349)) +. ( (#0.2384) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
11927 [((#4.0), x1, square_2t0);
11928 ((#4.0), x2, square_2t0);
11929 ((#4.0), x3, square_2t0);
11931 ((#4.0), x4, (square (#2.1773)));
11932 ((square (#2.1773)), x5, square_2t0);
11933 ((#4.0), x6, (square (#2.1773)))
11935 ( (tau_sigma_x x1 x2 x3 x4 x5 x6) >. ( (--. (#0.26303)) +. ( (#0.2384) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
11941 [((#4.0), x1, square_2t0);
11942 ((#4.0), x2, square_2t0);
11943 ((#4.0), x3, square_2t0);
11945 ((#4.0), x4, (square (#2.1773)));
11946 ((#4.0), x5, (square (#2.1773)));
11947 ((square (#2.1773)), x6, square_2t0)
11949 ( (tau_sigma_x x1 x2 x3 x4 x5 x6) >.
11950 ( (--. (#0.5565)) +. ( (#0.2384) *. ( (dih_x x1 x2 x3 x4 x5 x6) +. (dih2_x x1 x2 x3 x4 x5 x6)))))`;;
11956 [((#4.0), x1, square_2t0);
11957 ((#4.0), x2, square_2t0);
11958 ((#4.0), x3, square_2t0);
11960 ((#4.0), x4, (square (#2.1773)));
11961 ((#4.0), x5, (square (#2.1773)));
11962 ((#4.0), x6, (square (#2.1773)))
11964 ( (tau_sigma_x x1 x2 x3 x4 x5 x6) >.
11965 ( ( (--. (#2.0)) *. (#0.29349)) +. ( (#0.2384) *. ( (dih_x x1 x2 x3 x4 x5 x6) +. (dih2_x x1 x2 x3 x4 x5 x6)))))`;;
11971 [((#4.0), x1, square_2t0);
11972 ((#4.0), x2, square_2t0);
11973 ((#4.0), x3, square_2t0);
11975 ((#4.0), x4, (square (#2.1773)));
11976 ((#4.0), x5, (square (#2.1773)));
11977 ((#4.0), x6, (square (#2.1773)))
11979 ( (tau_sigma_x x1 x2 x3 x4 x5 x6) >.
11980 ( ( (--. (#3.0)) *. (#0.29349)) +.
11981 ( (#0.2384) *. ( (dih_x x1 x2 x3 x4 x5 x6) +. (dih2_x x1 x2 x3 x4 x5 x6) +.
11982 (dih3_x x1 x2 x3 x4 x5 x6)))))`;;
11987 J_550901847;J_559163627;J_571492944;J_471806843;J_610154063;
11988 J_466112442;J_904445624;];;
11993 [((#4.0), x1, square_2t0);
11994 ((#4.0), x2, square_2t0);
11995 ((#4.0), x3, square_2t0);
11997 ((square (#2.177303)), x4, square_2t0);
11998 ((#4.0), x5, square_2t0);
11999 ((#4.0), x6, square_2t0)
12001 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <. ( ( (#1.0) +. (--. (#0.48))) *. pt))`;;
12003 (* Added March 10, 2005. Requested by Lagarias for DCG *)
12004 (* Note to Google flyspeck group, March 10, 2005 *)
12005 (* moved 241241504_1 to inequality_spec.ml *)
12010 [((#4.0), x1, square_2t0);
12011 ((#4.0), x2, square_2t0);
12012 ((#4.0), x3, square_2t0);
12014 ((square (#2.177303)), x4, square_2t0);
12015 ((square (#2.177303)), x5, square_2t0);
12016 ((#4.0), x6, square_2t0)
12018 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <. ( ( (#1.0) +. ( (--. (#2.0)) *. (#0.48))) *. pt))`;;
12023 (* moved 82950290 to inequality_spec.ml *)
12030 [((#4.0), x1, square_2t0);
12031 ((#4.0), x2, square_2t0);
12032 ((#4.0), x3, square_2t0);
12034 ((#4.0), x4, (square (#2.177303)));
12035 ((square (#2.177303)), x5, square_2t0);
12036 ((#4.0), x6, (square (#2.177303)))
12038 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
12039 ( (#0.28365) +. ( (--. (#0.207045)) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
12044 [((#4.0), x1, square_2t0);
12045 ((#4.0), x2, square_2t0);
12046 ((#4.0), x3, square_2t0);
12048 ((#4.0), x4, (square (#2.177303)));
12049 ((#4.0), x5, (square (#2.177303)));
12050 ((square (#2.177303)), x6, square_2t0)
12052 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
12053 ( (#0.53852) +. ( (--. (#0.207045)) *. ( (dih_x x1 x2 x3 x4 x5 x6) +. (dih2_x x1 x2 x3 x4 x5 x6)))))`;;
12059 [((#4.0), x1, square_2t0);
12060 ((#4.0), x2, square_2t0);
12061 ((#4.0), x3, square_2t0);
12063 ((#4.0), x4, (square (#2.177303)));
12064 ((#4.0), x5, (square (#2.177303)));
12065 ((#4.0), x6, (square (#2.177303)))
12067 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
12068 ( (( --. ) (pt)) +. ( (#2.0) *. (#0.31023815)) +.
12069 ( (--. (#0.207045)) *. ( (dih_x x1 x2 x3 x4 x5 x6) +. (dih2_x x1 x2 x3 x4 x5 x6)))))`;;
12075 [((#4.0), x1, square_2t0);
12076 ((#4.0), x2, square_2t0);
12077 ((#4.0), x3, square_2t0);
12079 ((#4.0), x4, (square (#2.177303)));
12080 ((#4.0), x5, (square (#2.177303)));
12081 ((#4.0), x6, (square (#2.177303)))
12083 ( (sigma_qrtet_x x1 x2 x3 x4 x5 x6) <.
12084 ( ( (--. (#2.0)) *. pt) +. ( (#3.0) *. (#0.31023815)) +.
12085 ( (--. (#0.207045)) *. ( (dih_x x1 x2 x3 x4 x5 x6) +. (dih2_x x1 x2 x3 x4 x5 x6) +.
12086 (dih3_x x1 x2 x3 x4 x5 x6)))))`;;
12089 let C_129662166 = list_mk_conj [
12090 J_241241504;J_144820927;J_82950290;J_938408928;J_739415811;
12091 J_898558502;J_413792383;];;
12098 LOC: 2002 III, page 17.
12099 Section A.2 (Flat Quarters)
12106 [((#4.0), x1, square_2t0);
12107 ((#4.0), x2, square_2t0);
12108 ((#4.0), x3, square_2t0);
12110 (square_2t0, x4, (#8.0));
12111 ((#4.0), x5, square_2t0);
12112 ((#4.0), x6, square_2t0)
12115 ( (( --. ) (dih2_x x1 x2 x3 x4 x5 x6)) +. ( (#0.35) *. (sqrt x2)) +. ( (--. (#0.15)) *. (sqrt x1)) +.
12116 ( (--. (#0.15)) *. (sqrt x3)) +. ( (#0.7022) *. (sqrt x5)) +. ( (--. (#0.17)) *. (sqrt x4))) >. (--. (#0.0123)))`;;
12122 [((#4.0), x1, square_2t0);
12123 ((#4.0), x2, square_2t0);
12124 ((#4.0), x3, square_2t0);
12126 (square_2t0, x4, (#8.0));
12127 ((#4.0), x5, square_2t0);
12128 ((#4.0), x6, square_2t0)
12131 ( (( --. ) (dih3_x x1 x2 x3 x4 x5 x6)) +. ( (#0.35) *. (sqrt x3)) +. ( (--. (#0.15)) *. (sqrt x1)) +.
12132 ( (--. (#0.15)) *. (sqrt x2)) +. ( (#0.7022) *. (sqrt x6)) +. ( (--. (#0.17)) *. (sqrt x4))) >. (--. (#0.0123)))`;;
12138 [((#4.0), x1, square_2t0);
12139 ((#4.0), x2, square_2t0);
12140 ((#4.0), x3, square_2t0);
12142 (square_2t0, x4, (#8.0));
12143 ((#4.0), x5, square_2t0);
12144 ((#4.0), x6, square_2t0)
12147 ( (dih2_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.13)) *. (sqrt x2)) +. ( (#0.631) *. (sqrt x1)) +.
12148 ( (#0.31) *. (sqrt x3)) +. ( (--. (#0.58)) *. (sqrt x5)) +. ( (#0.413) *. (sqrt x4)) +. ( (#0.025) *. (sqrt x6))) >.
12155 [((#4.0), x1, square_2t0);
12156 ((#4.0), x2, square_2t0);
12157 ((#4.0), x3, square_2t0);
12159 (square_2t0, x4, (#8.0));
12160 ((#4.0), x5, square_2t0);
12161 ((#4.0), x6, square_2t0)
12164 ( (dih3_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.13)) *. (sqrt x3)) +. ( (#0.631) *. (sqrt x1)) +.
12165 ( (#0.31) *. (sqrt x2)) +. ( (--. (#0.58)) *. (sqrt x6)) +. ( (#0.413) *. (sqrt x4)) +. ( (#0.025) *. (sqrt x5))) >.
12172 [((#4.0), x1, square_2t0);
12173 ((#4.0), x2, square_2t0);
12174 ((#4.0), x3, square_2t0);
12176 (square_2t0, x4, (#8.0));
12177 ((#4.0), x5, square_2t0);
12178 ((#4.0), x6, square_2t0)
12181 ( (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) +. ( (#0.714) *. (sqrt x1)) +. ( (--. (#0.221)) *. (sqrt x2)) +.
12182 ( (--. (#0.221)) *. (sqrt x3)) +. ( (#0.92) *. (sqrt x4)) +. ( (--. (#0.221)) *. (sqrt x5)) +. ( (--. (#0.221)) *. (sqrt x6))) >.
12189 [((#4.0), x1, square_2t0);
12190 ((#4.0), x2, square_2t0);
12191 ((#4.0), x3, square_2t0);
12193 (square_2t0, x4, (#8.0));
12194 ((#4.0), x5, square_2t0);
12195 ((#4.0), x6, square_2t0)
12198 ( (dih_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.315)) *. (sqrt x1)) +. ( (#0.3972) *. (sqrt x2)) +.
12199 ( (#0.3972) *. (sqrt x3)) +. ( (--. (#0.715)) *. (sqrt x4)) +. ( (#0.3972) *. (sqrt x5)) +. ( (#0.3972) *. (sqrt x6))) >.
12206 [((#4.0), x1, square_2t0);
12207 ((#4.0), x2, square_2t0);
12208 ((#4.0), x3, square_2t0);
12210 (square_2t0, x4, (#8.0));
12211 ((#4.0), x5, square_2t0);
12212 ((#4.0), x6, square_2t0)
12215 ( (( --. ) (sol_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.187)) *. (sqrt x1)) +. ( (--. (#0.187)) *. (sqrt x2)) +.
12216 ( (--. (#0.187)) *. (sqrt x3)) +. ( (#0.1185) *. (sqrt x4)) +. ( (#0.479) *. (sqrt x5)) +. ( (#0.479) *. (sqrt x6))) >.
12224 [((#4.0), x1, square_2t0);
12225 ((#4.0), x2, square_2t0);
12226 ((#4.0), x3, square_2t0);
12228 (square_2t0, x4, (#8.0));
12229 ((#4.0), x5, square_2t0);
12230 ((#4.0), x6, square_2t0)
12233 ( (sol_x x1 x2 x3 x4 x5 x6) +. ( (#0.488) *. (sqrt x1)) +. ( (#0.488) *. (sqrt x2)) +.
12234 ( (#0.488) *. (sqrt x3)) +. ( (--. (#0.334)) *. (sqrt x5)) +. ( (--. (#0.334)) *. (sqrt x6))) >.
12241 [((#4.0), x1, square_2t0);
12242 ((#4.0), x2, square_2t0);
12243 ((#4.0), x3, square_2t0);
12245 (square_2t0, x4, (#8.0));
12246 ((#4.0), x5, square_2t0);
12247 ((#4.0), x6, square_2t0)
12250 ( (( --. ) (mu_flat_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.159)) *. (sqrt x1)) +. ( (--. (#0.081)) *. (sqrt x2)) +.
12251 ( (--. (#0.081)) *. (sqrt x3)) +. ( (--. (#0.133)) *. (sqrt x5)) +. ( (--. (#0.133)) *. (sqrt x6))) >.
12252 (--. (#1.17401)))`;;
12258 [((#4.0), x1, square_2t0);
12259 ((#4.0), x2, square_2t0);
12260 ((#4.0), x3, square_2t0);
12262 (square_2t0, x4, (#8.0));
12263 ((#4.0), x5, square_2t0);
12264 ((#4.0), x6, square_2t0)
12267 (mu_flat_x x1 x2 x3 x4 x5 x6) <.
12268 ( ( (--. (#0.419351)) *. (sol_x x1 x2 x3 x4 x5 x6)) +. (#0.1448) +.
12269 ( (#0.0436) *. ( (sqrt x5) +. (sqrt x6) +. (--. (#4.0)))) +. ( (#0.079431) *. (dih_x x1 x2 x3 x4 x5 x6))))`;;
12275 [((#4.0), x1, square_2t0);
12276 ((#4.0), x2, square_2t0);
12277 ((#4.0), x3, square_2t0);
12279 (square_2t0, x4, (#8.0));
12280 ((#4.0), x5, square_2t0);
12281 ((#4.0), x6, square_2t0)
12284 (mu_flat_x x1 x2 x3 x4 x5 x6) <.
12285 ( (#0.000001) +. ( (--. (#0.197)) *. ( (sqrt x4) +. (sqrt x5) +. (sqrt x6) +. ( (--. (#2.0)) *. (sqrt (#2.0))) +. (--. (#4.0))))))`;;
12290 LOC: 2002 III, page 17-18
12291 Appendix 1 (Some final cases)
12292 Section A3 (upright quarters)
12298 [(square_2t0, x1, (#8.0));
12299 ((#4.0), x2, square_2t0);
12300 ((#4.0), x3, square_2t0);
12302 ((#4.0), x4, square_2t0);
12303 ((#4.0), x5, square_2t0);
12304 ((#4.0), x6, square_2t0)
12307 ( (dih_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.636)) *. (sqrt x1)) +. ( (#0.462) *. (sqrt x2)) +.
12308 ( (#0.462) *. (sqrt x3)) +. ( (--. (#0.82)) *. (sqrt x4)) +. ( (#0.462) *. (sqrt x5)) +. ( (#0.462) *. (sqrt x6))) >.
12315 [(square_2t0, x1, (#8.0));
12316 ((#4.0), x2, square_2t0);
12317 ((#4.0), x3, square_2t0);
12319 ((#4.0), x4, square_2t0);
12320 ((#4.0), x5, square_2t0);
12321 ((#4.0), x6, square_2t0)
12324 ( (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) +. ( (#0.55) *. (sqrt x1)) +. ( (--. (#0.214)) *. (sqrt x2)) +.
12325 ( (--. (#0.214)) *. (sqrt x3)) +. ( (#1.24) *. (sqrt x4)) +. ( (--. (#0.214)) *. (sqrt x5)) +. ( (--. (#0.214)) *. (sqrt x6))) >.
12332 [(square_2t0, x1, (#8.0));
12333 ((#4.0), x2, square_2t0);
12334 ((#4.0), x3, square_2t0);
12336 ((#4.0), x4, square_2t0);
12337 ((#4.0), x5, square_2t0);
12338 ((#4.0), x6, square_2t0)
12341 ( (dih2_x x1 x2 x3 x4 x5 x6) +. ( (#0.4) *. (sqrt x1)) +. ( (--. (#0.15)) *. (sqrt x2)) +.
12342 ( (#0.09) *. (sqrt x3)) +. ( (#0.631) *. (sqrt x4)) +. ( (--. (#0.57)) *. (sqrt x5)) +. ( (#0.23) *. (sqrt x6))) >.
12349 [(square_2t0, x1, (#8.0));
12350 ((#4.0), x2, square_2t0);
12351 ((#4.0), x3, square_2t0);
12353 ((#4.0), x4, square_2t0);
12354 ((#4.0), x5, square_2t0);
12355 ((#4.0), x6, square_2t0)
12358 ( (( --. ) (dih2_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.454)) *. (sqrt x1)) +. ( (#0.34) *. (sqrt x2)) +.
12359 ( (#0.154) *. (sqrt x3)) +. ( (--. (#0.346)) *. (sqrt x4)) +. ( (#0.805) *. (sqrt x5))) >.
12360 (--. (#0.3429)))`;;
12367 [(square_2t0, x1, (#8.0));
12368 ((#4.0), x2, square_2t0);
12369 ((#4.0), x3, square_2t0);
12371 ((#4.0), x4, square_2t0);
12372 ((#4.0), x5, square_2t0);
12373 ((#4.0), x6, square_2t0)
12376 ( (dih3_x x1 x2 x3 x4 x5 x6) +. ( (#0.4) *. (sqrt x1)) +. ( (--. (#0.15)) *. (sqrt x3)) +.
12377 ( (#0.09) *. (sqrt x2)) +. ( (#0.631) *. (sqrt x4)) +. ( (--. (#0.57)) *. (sqrt x6)) +. ( (#0.23) *. (sqrt x5))) >.
12385 [(square_2t0, x1, (#8.0));
12386 ((#4.0), x2, square_2t0);
12387 ((#4.0), x3, square_2t0);
12389 ((#4.0), x4, square_2t0);
12390 ((#4.0), x5, square_2t0);
12391 ((#4.0), x6, square_2t0)
12394 ( (( --. ) (dih3_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.454)) *. (sqrt x1)) +. ( (#0.34) *. (sqrt x3)) +.
12395 ( (#0.154) *. (sqrt x2)) +. ( (--. (#0.346)) *. (sqrt x4)) +. ( (#0.805) *. (sqrt x6))) >.
12396 (--. (#0.3429)))`;;
12402 [(square_2t0, x1, (#8.0));
12403 ((#4.0), x2, square_2t0);
12404 ((#4.0), x3, square_2t0);
12406 ((#4.0), x4, square_2t0);
12407 ((#4.0), x5, square_2t0);
12408 ((#4.0), x6, square_2t0)
12411 ( (sol_x x1 x2 x3 x4 x5 x6) +. ( (#0.065) *. (sqrt x2)) +. ( (#0.065) *. (sqrt x3)) +.
12412 ( (#0.061) *. (sqrt x4)) +. ( (--. (#0.115)) *. (sqrt x5)) +. ( (--. (#0.115)) *. (sqrt x6))) >.
12419 [(square_2t0, x1, (#8.0));
12420 ((#4.0), x2, square_2t0);
12421 ((#4.0), x3, square_2t0);
12423 ((#4.0), x4, square_2t0);
12424 ((#4.0), x5, square_2t0);
12425 ((#4.0), x6, square_2t0)
12428 ( (( --. ) (sol_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.239)) *. (sqrt x1)) +. ( (--. (#0.03)) *. (sqrt x2)) +.
12429 ( (--. (#0.03)) *. (sqrt x3)) +. ( (#0.12) *. (sqrt x4)) +. ( (#0.325) *. (sqrt x5)) +. ( (#0.325) *. (sqrt x6))) >.
12436 [(square_2t0, x1, (#8.0));
12437 ((#4.0), x2, square_2t0);
12438 ((#4.0), x3, square_2t0);
12439 ((#4.0), x4, square_2t0);
12440 ((#4.0), x5, square_2t0);
12441 ((#4.0), x6, square_2t0)
12444 ( (( --. ) (octa_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.054)) *. (sqrt x2)) +. ( (--. (#0.054)) *. (sqrt x3)) +.
12445 ( (--. (#0.083)) *. (sqrt x4)) +. ( (--. (#0.054)) *. (sqrt x5)) +. ( (--. (#0.054)) *. (sqrt x6))) >.
12446 (--. (#0.59834)))`;;
12452 [(square_2t0, x1, (#8.0));
12453 ((#4.0), x2, square_2t0);
12454 ((#4.0), x3, square_2t0);
12456 ((#4.0), x4, square_2t0);
12457 ((#4.0), x5, square_2t0);
12458 ((#4.0), x6, square_2t0)
12461 (octa_x x1 x2 x3 x4 x5 x6) <.
12462 ( ( (--. (#0.419351)) *. (sol_x x1 x2 x3 x4 x5 x6)) +. ( (#0.079431) *. (dih2_x x1 x2 x3 x4 x5 x6)) +.
12463 (#0.06904) +. ( (--. (#0.0846)) *. ( (sqrt x1) +. (--. (#2.8))))))`;;
12469 [(square_2t0, x1, (#8.0));
12470 ((#4.0), x2, (square (#2.13)));
12471 ((#4.0), x3, (square (#2.13)));
12473 ((#4.0), x4, square_2t0);
12474 ((#4.0), x5, square_2t0);
12475 ((#4.0), x6, square_2t0)
12478 (octa_x x1 x2 x3 x4 x5 x6) <.
12479 ( ( (#0.07) *. ( (sqrt x1) +. (--. (#2.51)))) +. ( (--. (#0.133)) *. ( (sqrt x2) +. (sqrt x3) +. (sqrt x5) +. (sqrt x6) +. (--. (#8.0)))) +.
12480 ( (#0.135) *. ( (sqrt x4) +. (--. (#2.0))))))`;;
12485 LOC: 2002 III, page 18.
12486 Appendix 1. (Some final cases)
12487 Section A4 (Truncated quad clusters)
12494 [((#4.0), x1, square_2t0);
12495 ((#4.0), x2, square_2t0);
12496 ((#4.0), x3, square_2t0);
12498 ((#8.0), x4, square_4t0);
12499 ((#4.0), x5, square_2t0);
12500 ((#4.0), x6, square_2t0)
12503 ( (dih_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.372)) *. (sqrt x1)) +. ( (#0.465) *. (sqrt x2)) +.
12504 ( (#0.465) *. (sqrt x3)) +. ( (#0.465) *. (sqrt x5)) +. ( (#0.465) *. (sqrt x6))) >.
12511 [((#4.0), x1, (square (#2.26)));
12512 ((#4.0), x2, (square (#2.26)));
12513 ((#4.0), x3, (square (#2.26)));
12515 ((#8.0), x4, square_4t0);
12516 ((#4.0), x5, square_2t0);
12517 ((#4.0), x6, square_2t0)
12520 ( ( (( --. ) (vor_0_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.06)) *. (sqrt x2)) +. ( (--. (#0.06)) *. (sqrt x3)) +.
12521 ( (--. (#0.185)) *. (sqrt x5)) +. ( (--. (#0.185)) *. (sqrt x6))) >. (--. (#0.9978))) \/
12522 ( (dih_x x1 x2 x3 x4 x5 x6) >=. (#2.12)))`;;
12528 [((#4.0), x1, (square (#2.26)));
12529 ((#4.0), x2, (square (#2.26)));
12530 ((#4.0), x3, (square (#2.26)));
12532 ((#8.0), x4, square_4t0);
12533 ((#4.0), x5, square_2t0);
12534 ((#4.0), x6, square_2t0)
12537 ( ( (( --. ) (vor_0_x x1 x2 x3 x4 x5 x6)) +. ( (#0.419351) *. (sol_x x1 x2 x3 x4 x5 x6))) <. (#0.3072)) \/
12538 ( (dih_x x1 x2 x3 x4 x5 x6) >=. (#2.12)))`;;
12544 [((#4.0), x1, square_2t0);
12545 ((#4.0), x2, square_2t0);
12546 ((#4.0), x6, square_2t0)
12548 ( ( (quo_x x1 x2 x6) +. ( (#0.00758) *. (sqrt x1)) +. ( (#0.0115) *. (sqrt x2)) +. ( (#0.0115) *. (sqrt x6))) >.
12554 Handwritten in as new inequality
12558 [((#4.0), x1, square_2t0);
12559 ((#4.0), x2, square_2t0);
12560 ((#4.0), x6, square_2t0)
12562 ( (quo_x x1 x2 x6) >=. (#0.0))`;;
12569 LOC: 2002 III, page 19.
12570 Appendix 1. (Some final cases)
12571 Section A6 (Quasi-regular tetrahedra)
12577 [((#4.0), x1, square_2t0);
12578 ((#4.0), x2, square_2t0);
12579 ((#4.0), x3, square_2t0);
12581 ((#4.0), x4, square_2t0);
12582 ((#4.0), x5, square_2t0);
12583 ((#4.0), x6, square_2t0)
12587 ( (sol_x x1 x2 x3 x4 x5 x6) +. ( (#0.377076) *. (sqrt x1)) +. ( (#0.377076) *. (sqrt x2)) +.
12588 ( (#0.377076) *. (sqrt x3)) +. ( (--. (#0.221)) *. (sqrt x4)) +. ( (--. (#0.221)) *. (sqrt x5)) +. ( (--. (#0.221)) *. (sqrt x6))) >.
12590 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) >. (#6.25)))`;;
12596 [((#4.0), x1, square_2t0);
12597 ((#4.0), x2, square_2t0);
12598 ((#4.0), x3, square_2t0);
12600 ((#4.0), x4, square_2t0);
12601 ((#4.0), x5, square_2t0);
12602 ((#4.0), x6, square_2t0)
12606 ( ( (#0.221) *. (sqrt x4)) +. ( (#0.221) *. (sqrt x5)) +. ( (#0.221) *. (sqrt x6)) +. (( --. ) (sol_x x1 x2 x3 x4 x5 x6))) >.
12608 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) >. (#6.25)))`;;
12614 [((#4.0), x1, square_2t0);
12615 ((#4.0), x2, square_2t0);
12616 ((#4.0), x3, square_2t0);
12618 ((#4.0), x4, square_2t0);
12619 ((#4.0), x5, square_2t0);
12620 ((#4.0), x6, square_2t0)
12624 ( (dih_x x1 x2 x3 x4 x5 x6) +. ( (#0.34) *. (sqrt x2)) +. ( (#0.34) *. (sqrt x3)) +.
12625 ( (--. (#0.689)) *. (sqrt x4)) +. ( (#0.27) *. (sqrt x5)) +. ( (#0.27) *. (sqrt x6))) >.
12627 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) >. (#6.25)))`;;
12633 [((#4.0), x1, square_2t0);
12634 ((#4.0), x2, square_2t0);
12635 ((#4.0), x3, square_2t0);
12637 ((#4.0), x4, square_2t0);
12638 ((#4.0), x5, square_2t0);
12639 ((#4.0), x6, square_2t0)
12643 ( (( --. ) (dih_x x1 x2 x3 x4 x5 x6)) +. ( (#0.498) *. (sqrt x1)) +. ( (#0.731) *. (sqrt x4)) +.
12644 ( (--. (#0.212)) *. (sqrt x5)) +. ( (--. (#0.212)) *. (sqrt x6))) >.
12646 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) >. (#6.25)))`;;
12653 [((#4.0), x1, square_2t0);
12654 ((#4.0), x2, square_2t0);
12655 ((#4.0), x3, square_2t0);
12657 ((#4.0), x4, square_2t0);
12658 ((#4.0), x5, square_2t0);
12659 ((#4.0), x6, square_2t0)
12663 ( (( --. ) (sigma_qrtet_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.109)) *. (sqrt x1)) +. ( (--. (#0.109)) *. (sqrt x2)) +.
12664 ( (--. (#0.109)) *. (sqrt x3)) +. ( (--. (#0.14135)) *. (sqrt x4)) +. ( (--. (#0.14135)) *. (sqrt x5)) +. ( (--. (#0.14135)) *. (sqrt x6))) >.
12665 (--. (#1.5574737))) \/
12666 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) >. (#6.25)))`;;
12673 [((#4.0), x1, square_2t0);
12674 ((#4.0), x2, square_2t0);
12675 ((#4.0), x3, square_2t0);
12677 ((#4.0), x4, square_2t0);
12678 ((#4.0), x5, square_2t0);
12679 ((#4.0), x6, square_2t0)
12683 ( (( --. ) (sigma_qrtet_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.419351)) *. (sol_x x1 x2 x3 x4 x5 x6)) +.
12684 ( (--. (#0.2)) *. (sqrt x1)) +. ( (--. (#0.2)) *. (sqrt x2)) +. ( (--. (#0.2)) *. (sqrt x3)) +.
12685 ( (--. (#0.048)) *. (sqrt x4)) +. ( (--. (#0.048)) *. (sqrt x5)) +. ( (--. (#0.048)) *. (sqrt x6))) >.
12686 (--. (#1.77465))) \/
12687 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) >. (#6.25)))`;;
12693 [((#4.0), x1, square_2t0);
12694 ((#4.0), x2, square_2t0);
12695 ((#4.0), x3, square_2t0);
12697 ((#4.0), x4, square_2t0);
12698 ((#4.0), x5, square_2t0);
12699 ((#4.0), x6, square_2t0)
12703 ( (tau_sigma_x x1 x2 x3 x4 x5 x6) +. ( (--. (#0.0845696)) *. (sqrt x1)) +. ( (--. (#0.0845696)) *. (sqrt x2)) +.
12704 ( (--. (#0.0845696)) *. (sqrt x3)) +. ( (--. (#0.163)) *. (sqrt x4)) +. ( (--. (#0.163)) *. (sqrt x5)) +. ( (--. (#0.163)) *. (sqrt x6))) >.
12705 (--. (#1.48542))) \/
12706 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) >. (#6.25)))`;;
12712 [((#4.0), x1, (square (#2.13)));
12713 ((#4.0), x2, (square (#2.13)));
12714 ((#4.0), x3, (square (#2.13)));
12716 ((#4.0), x4, square_2t0);
12717 ((#4.0), x5, square_2t0);
12718 ((#4.0), x6, square_2t0)
12722 ( (dih_x x1 x2 x3 x4 x5 x6) +. ( (#0.27) *. (sqrt x2)) +. ( (#0.27) *. (sqrt x3)) +.
12723 ( (--. (#0.689)) *. (sqrt x4)) +. ( (#0.27) *. (sqrt x5)) +. ( (#0.27) *. (sqrt x6))) >.
12725 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) >. (#6.25)))`;;
12730 Bound: 0.0207472140662
12732 Point: [4.3332764986, 4.0214270778, 4.12912710387, 5.03818306425, 5.36790850639, 4.93072243755]
12734 yy = {4.3332764986, 4.0214270778, 4.12912710387, 5.03818306425, 5.36790850639, 4.93072243755}//Sqrt
12736 cstr1 = ( yy[[4]]+yy[[5]]+yy[[6]] > 6.25 )
12738 cstr2 = ( -(Gamma @@ yy) - 0.14135 (Plus @@ yy) > -1.7515737 )
12740 (* both constraints are satisfied. This is not a counterexample. It is not
12741 even close to being a counterexample. Why does a question even come up? *)
12745 [((#4.0), x1, (square (#2.13)));
12746 ((#4.0), x2, (square (#2.13)));
12747 ((#4.0), x3, (square (#2.13)));
12749 ((#4.0), x4, square_2t0);
12750 ((#4.0), x5, square_2t0);
12751 ((#4.0), x6, square_2t0)
12755 ( (( --. ) (sigma_qrtet_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.14135)) *. (sqrt x1)) +. ( (--. (#0.14135)) *. (sqrt x2)) +.
12756 ( (--. (#0.14135)) *. (sqrt x3)) +. ( (--. (#0.14135)) *. (sqrt x4)) +. ( (--. (#0.14135)) *. (sqrt x5)) +. ( (--. (#0.14135)) *. (sqrt x6))) >.
12757 (--. (#1.7515737))) \/
12758 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) >. (#6.25)))`;;
12765 [((#4.0), x1, square_2t0);
12766 ((#4.0), x2, square_2t0);
12767 ((#4.0), x3, square_2t0);
12769 ((#4.0), x4, square_2t0);
12770 ((#4.0), x5, square_2t0);
12771 ((#4.0), x6, square_2t0)
12775 ( (sol_x x1 x2 x3 x4 x5 x6) +. ( (#0.378) *. (sqrt x1)) +. ( (#0.378) *. (sqrt x2)) +.
12776 ( (#0.378) *. (sqrt x3)) +. ( (--. (#0.1781)) *. (sqrt x4)) +. ( (--. (#0.1781)) *. (sqrt x5)) +. ( (--. (#0.1781)) *. (sqrt x6))) >.
12778 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) <. (#6.25)))`;;
12785 [((#4.0), x1, square_2t0);
12786 ((#4.0), x2, square_2t0);
12787 ((#4.0), x3, square_2t0);
12789 ((#4.0), x4, square_2t0);
12790 ((#4.0), x5, square_2t0);
12791 ((#4.0), x6, square_2t0)
12795 ( (( --. ) (sol_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.171)) *. (sqrt x1)) +. ( (--. (#0.171)) *. (sqrt x2)) +.
12796 ( (--. (#0.171)) *. (sqrt x3)) +. ( (#0.3405) *. (sqrt x4)) +. ( (#0.3405) *. (sqrt x5)) +. ( (#0.3405) *. (sqrt x6))) >.
12798 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) <. (#6.25)))`;;
12804 [((#4.0), x1, square_2t0);
12805 ((#4.0), x2, square_2t0);
12806 ((#4.0), x3, square_2t0);
12808 ((#4.0), x4, square_2t0);
12809 ((#4.0), x5, square_2t0);
12810 ((#4.0), x6, square_2t0)
12814 ( (( --. ) (sigma_qrtet_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.1208)) *. (sqrt x1)) +. ( (--. (#0.1208)) *. (sqrt x2)) +.
12815 ( (--. (#0.1208)) *. (sqrt x3)) +. ( (--. (#0.0781)) *. (sqrt x4)) +. ( (--. (#0.0781)) *. (sqrt x5)) +. ( (--. (#0.0781)) *. (sqrt x6))) >.
12816 (--. (#1.2436))) \/
12817 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) <. (#6.25)))`;;
12823 [((#4.0), x1, square_2t0);
12824 ((#4.0), x2, square_2t0);
12825 ((#4.0), x3, square_2t0);
12827 ((#4.0), x4, square_2t0);
12828 ((#4.0), x5, square_2t0);
12829 ((#4.0), x6, square_2t0)
12833 ( (( --. ) (sigma_qrtet_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.419351)) *. (sol_x x1 x2 x3 x4 x5 x6)) +.
12834 ( (--. (#0.2)) *. (sqrt x1)) +. ( (--. (#0.2)) *. (sqrt x2)) +. ( (--. (#0.2)) *. (sqrt x3)) +.
12835 ( (#0.0106) *. (sqrt x4)) +. ( (#0.0106) *. (sqrt x5)) +. ( (#0.0106) *. (sqrt x6))) >.
12836 (--. (#1.40816))) \/
12837 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) <. (#6.25)))`;;
12844 [((#4.0), x1, (square (#2.13)));
12845 ((#4.0), x2, (square (#2.13)));
12846 ((#4.0), x3, (square (#2.13)));
12848 ((#4.0), x4, square_2t0);
12849 ((#4.0), x5, square_2t0);
12850 ((#4.0), x6, square_2t0)
12854 ( (sol_x x1 x2 x3 x4 x5 x6) +. ( (#0.356) *. (sqrt x1)) +. ( (#0.356) *. (sqrt x2)) +. ( (#0.356) *. (sqrt x3)) +.
12855 ( (--. (#0.1781)) *. (sqrt x4)) +. ( (--. (#0.1781)) *. (sqrt x5)) +. ( (--. (#0.1781)) *. (sqrt x6))) >.
12857 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) <. (#6.25)))`;;
12861 (* CCC Added 6.13 constraint on 2/1/2008.
12862 Bound: 0.0071026022964
12864 Point: [4.53689999999, 4.53689999999, 4.53689999999, 4.34027778215, 4.34027782266, 4.34027772851]
12866 yy = {y1,y2,y3,y4,y5,y6}={4.53689999999, 4.53689999999, 4.53689999999, 4.34027778215, 4.34027782266, 4.34027772851}//Sqrt;
12868 cnstr1 = (y4 + y5+y6 < 6.25) (* lands right at 6.25 *)
12870 constr2 = ( -(Solid @@ yy) - 0.254 (y1+y2+y3) + 0.3405 (y4+y5+y6) > -0.008855)
12873 (* interval verification in part3.cc (numbered as 465988688)
12874 Notes on interval verification.
12875 It uses constant -0.61298 + 0.3405 6.25 -0.254 6 = -0.008855.
12876 F is the main inequality.
12877 G is the y4+y5+y6 < 6.25 constraint.
12878 H is the inequality 6.13 < y1 + y2 +y3. H is not stated in SPVI2002.
12879 It seems to have been a constraint of the original inequality and then left
12880 out of the writeup. This explains the difference.
12881 There is one more inequality J that is a consequence of F, hence redundant.
12882 Note added to dcg_errata, adding the precondition.
12887 [((#4.0), x1, (square (#2.13)));
12888 ((#4.0), x2, (square (#2.13)));
12889 ((#4.0), x3, (square (#2.13)));
12891 ((#4.0), x4, square_2t0);
12892 ((#4.0), x5, square_2t0);
12893 ((#4.0), x6, square_2t0)
12897 ( (( --. ) (sol_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.254)) *. (sqrt x1)) +. ( (--. (#0.254)) *. (sqrt x2)) +. ( (--. (#0.254)) *. (sqrt x3)) +.
12898 ( (#0.3405) *. (sqrt x4)) +. ( (#0.3405) *. (sqrt x5)) +. ( (#0.3405) *. (sqrt x6))) >.
12899 (--. (#0.008855))) \/
12900 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) <. (#6.25)) \/
12901 ( ( (sqrt x1) +. (sqrt x2) +. (sqrt x3)) >. (#6.13)))`;;
12908 [((#4.0), x1, (square (#2.13)));
12909 ((#4.0), x2, (square (#2.13)));
12910 ((#4.0), x3, (square (#2.13)));
12912 ((#4.0), x4, square_2t0);
12913 ((#4.0), x5, square_2t0);
12914 ((#4.0), x6, square_2t0)
12918 ( (( --. ) (sigma_qrtet_x x1 x2 x3 x4 x5 x6)) +. ( (--. (#0.167)) *. (sqrt x1)) +. ( (--. (#0.167)) *. (sqrt x2)) +. ( (--. (#0.167)) *. (sqrt x3)) +.
12919 ( (--. (#0.0781)) *. (sqrt x4)) +. ( (--. (#0.0781)) *. (sqrt x5)) +. ( (--. (#0.0781)) *. (sqrt x6))) >.
12920 (--. (#1.51017))) \/
12921 ( ( (sqrt x4) +. (sqrt x5) +. (sqrt x6)) <. (#6.25)))`;;
12926 LOC: 2002 III, page 20.
12927 Appendix 1. (Some final Cases)
12928 Section A8 (Final cases)
12934 [((#4.0), x1, (square (#2.13)));
12935 ((#4.0), x2, (square (#2.13)));
12936 ((#4.0), x3, (square (#2.13)));
12938 ((square (#2.93)), x4, square_4t0);
12939 ((#4.0), x5, square_2t0);
12940 ((#4.0), x6, square_2t0)
12943 ( (dih_x x1 x2 x3 x4 x5 x6) >. (#1.694)) \/
12944 ( ( (sqrt x2) +. (sqrt x3) +. (sqrt x5) +. (sqrt x6)) >. (#8.709)))`;;
12951 [((#4.0), x1, (square (#2.13)));
12952 ((#4.0), x2, (square (#2.13)));
12953 ((#4.0), x3, (square (#2.13)));
12955 ((#8.0), x4, (square (#2.93)));
12956 ((#4.0), x5, square_2t0);
12957 ((#4.0), x6, square_2t0)
12959 ( ( (dih2_x x1 x2 x3 x4 x5 x6) +. ( (#0.59) *. (sqrt x1)) +. ( (#0.1) *. (sqrt x2)) +. ( (#0.1) *. (sqrt x3)) +.
12960 ( (#0.55) *. (sqrt x4)) +. ( (--. (#0.6)) *. (sqrt x5)) +. ( (--. (#0.12)) *. (sqrt x6))) >. (#2.6506))`;;
12966 [((#4.0), x1, (square (#2.13)));
12967 ((#4.0), x2, (square (#2.13)));
12968 ((#4.0), x3, (square (#2.13)));
12970 ((#8.0), x4, (square (#2.93)));
12971 ((#4.0), x5, square_2t0);
12972 ((#4.0), x6, square_2t0)
12974 ( ( (dih2_x x1 x2 x3 x4 x5 x6) +. ( (#0.35) *. (sqrt x1)) +. ( (--. (#0.24)) *. (sqrt x2)) +. ( (#0.05) *. (sqrt x3)) +.
12975 ( (#0.35) *. (sqrt x4)) +. ( (--. (#0.72)) *. (sqrt x5)) +. ( (--. (#0.18)) *. (sqrt x6))) <. (#0.47))`;;
12980 LOC: DCG II, page 147 (published DCG pages).
12981 Cases (8) (9) (10) (11)
12982 Used in Formulation
12984 CCC Fixed circumradius constraints 2/1/2008
12986 Bound: 0.00257586721418
12988 Point: [8, 3.99999999999, 6.30009999999, 3.99999999999, 8, 4]
12990 yy = {8, 3.99999999999, 6.30009999999, 3.99999999999, 8, 4}//Sqrt
12996 (* moved 629256313 to inequality_spec.ml *)
12999 (* eta_x constraint fixed 2/1/2008 *)
13001 (* moved 917032944 to inequality_spec.ml *)
13004 (* eta_x constraint fixed 2/1/2008 *)
13006 (* moved 738318844 to inequality_spec.ml *)
13009 (* eta_x constraint fixed 2/1/2008 *)
13011 (* moved 587618947 to inequality_spec.ml *)
13017 LOC: DCG Sphere Packing II, page 147, Calc 4.5.1.
13019 Note case of equality is equality five (#4.0) and x4=(#8.0).
13020 In the following inequality, we need that this is the unique case
13024 (* moved 346093004 to inequality_spec.ml *)
13029 (* I, SPI-1997 Lemma 9.17 *)
13033 [((#4.0), x1, square_2t0);
13034 ((#4.0), x2, square_2t0);
13035 ((#4.0), x3, square_2t0);
13036 ((#4.0), x4, square_2t0);
13037 ((#4.0), x5, square_2t0);
13038 ((#4.0), x6, square_2t0)
13040 (((vor_analytic_x x1 x2 x3 x4 x5 x6) < --((#1.8))*pt) \/
13041 (rad2_x x1 x2 x3 x4 x5 x6 <. (#1.9881)))`;;
13047 LOC: 2002 Form, Appendix 1, page 19
13052 LOC: 2002 Form, Appendix 1, page 19
13057 (* moved 5901405 to inequality_spec.ml *)
13062 LOC: 2002 Form, Appendix 1, page 19
13064 We need that equality implies that x1=8 and the other edges are 4.0.
13066 (* moved 40003553 to inequality_spec.ml *)
13071 LOC: 2002 Form, Appendix 1, page 19
13074 (* moved 522528841 to inequality_spec.ml *)
13079 LOC: 2002 Form, Appendix 1, page 19
13082 (* moved 892806084 to inequality_spec.ml *)
13087 LOC: 2002 Form, Appendix 1, page 20
13088 2002_Formulation_4.7.1:
13090 Corrected 2/1/2008, mu_flat_x -> mu_upright_x
13093 Bound: 0.394287252586
13095 Point: [7.99999999999, 3.99999999999, 6.30009999999, 6.30009999999, 3.99999999999, 4.00000002705]
13097 yy = {y1,y2,y3,y4,y5,y6}={7.99999999999, 3.99999999999, 6.30009999999, 6.30009999999, 3.99999999999, 4.00000002705}//Sqrt;
13099 CCC 3/10/2008. I had the domain swapped on x1 x4. I think it is OK now.
13100 this still fails in almost every case
13105 Functions : vor_analytic_x[x1, x2, x3, x4, x5, x6] + vor_analytic_x_flipped[x1, x2, x3, x4, x5, x6] + (crown[(sqrt x1 / 2.0)] * 1.0) + (crown[(sqrt x1 / 2.0)] * ((~ * dih_x[x1, x2, x3, x4, x5, x6]) / pi)) + 2.0 anc[sqrt x1, sqrt x2, sqrt x6] + ~ vor_0_x[x1, x2, x3, x4, x5, x6] + ~ vor_0_x_flipped[x1, x2, x3, x4, x5, x6]
13106 ~sqrt2 + eta_x[x2, x3, x4]
13107 ~sqrt2 + eta_x[x1, x5, x4]
13115 Bound: 0.278416202455
13117 Point: [7.50977085644, 4.00000080978, 5.91871675372, 4.00003052831, 5.91874664244, 4.00001152248]
13119 eta_x are near sqrt 2
13122 (* moved 554253147 to inequality_spec.ml *)
13127 LOC: 2002 Form, Appendix 1, page 20
13128 2002_Formulation_4.7.2:
13131 crown[Sqrt[2.575]] --> 0
13134 (* moved 906566422 to inequality_spec.ml *)
13139 LOC: 2002 Form, Appendix 1, page 20
13140 2002_Formulation_4.7.3:
13142 (* moved 703457064 to inequality_spec.ml *)
13148 LOC: 2002 Form, Appendix 1, page 20
13149 2002_Formulation_4.7.4
13151 (* moved 175514843 to inequality_spec.ml *)
13156 LOC: 2002 Form, Appendix 1, page 20
13157 2002_Formulation_4.7.5
13159 (* moved 855677395 to inequality_spec.ml *)
13164 (* ****************************************************** *)
13165 (* FERGUSON'S THESIS INEQUALITIES *)
13168 (* LOC: DCG 2006, V, page 197. Calc 17.4.1.1. *)
13170 (* verification uses dimension reduction *)
13173 [((#4.0),x1,square_2t0);
13174 ((#4.0),x2,square_2t0);
13175 ((#4.0),x3,square_2t0);
13176 ((#4.0),x4,square_2t0);
13177 ((#4.0),x5,square_2t0);
13178 ((#4.0),x6,square_2t0)
13180 ( (gamma_x x1 x2 x3 x4 x5 x6 < pp_a1 * dih_x x1 x2 x3 x4 x5 x6 - pp_a2) \/
13181 (gamma_x x1 x2 x3 x4 x5 x6 < -- (#0.52) * pt))`;;
13184 (* LOC: DCG 2006, V, page 197. Calc 17.4.1.2. *)
13186 (* verification uses dimension reduction *)
13189 [((#4.0),x1,square_2t0);
13190 ((#4.0),x2,square_2t0);
13191 ((#4.0),x3,square_2t0);
13192 ((#4.0),x4,square_2t0);
13193 ((#4.0),x5,square_2t0);
13194 ((#4.0),x6,square_2t0)
13196 ( (gamma_x x1 x2 x3 x4 x5 x6 < pp_a1 * dih_x x1 x2 x3 x4 x5 x6 + (#3.48 * pt) - (#2.0 * pi * pp_a1) + (#4.0 * pp_a2)) \/
13197 (gamma_x x1 x2 x3 x4 x5 x6 < -- (#0.52) * pt) \/
13198 (dih_x x1 x2 x3 x4 x5 x6 < pp_d0))`;;
13200 (* LOC: DCG 2006, V, page 197. Calc 17.4.1.3. *)
13202 (* verification uses dimension reduction *)
13205 [((#4.0),x1,square_2t0);
13206 ((#4.0),x2,square_2t0);
13207 ((#4.0),x3,square_2t0);
13208 ((#4.0),x4,square_2t0);
13209 ((#4.0),x5,square_2t0);
13210 ((#4.0),x6,square_2t0)
13212 ( (gamma_x x1 x2 x3 x4 x5 x6 + pp_m * sol_x x1 x2 x3 x4 x5 x6 +
13213 pp_a * (dih_x x1 x2 x3 x4 x5 x6 - #2.0 * pi / #5.0) < pp_bc) \/
13214 (gamma_x x1 x2 x3 x4 x5 x6 < -- (#0.52) * pt) \/
13215 (dih_x x1 x2 x3 x4 x5 x6 > pp_d0))`;;
13218 (* LOC: DCG 2006, V, page 198. Calc 17.4.2.1. *)
13220 (* verification uses dimension reduction. See note on Calc 17.4.2.2
13222 CCC typo fixed pp_bc -> pp_b
13223 Bound: 0.119559830004
13225 Point: [4.00000445799, 4.00000445799, 4.00000286459, 4.00004119188, 4.00004119188, 7.99987944373]
13232 [((#4.0),x1,square_2t0);
13233 ((#4.0),x2,square_2t0);
13234 ((#4.0),x3,square_2t0);
13235 ((#4.0),x4,square_2t0);
13236 ((#4.0),x5,square_2t0);
13237 (square_2t0,x6,(#8.0))
13239 ( (gamma_x x1 x2 x3 x4 x5 x6 + pp_m * sol_x x1 x2 x3 x4 x5 x6 < pp_b / (#2.0)) \/
13240 (eta_x x1 x2 x6 > sqrt2) \/
13241 (eta_x x4 x5 x6 > sqrt2) \/
13242 (gamma_x x1 x2 x3 x4 x5 x6 < -- (#1.04)* pt))`;;
13244 (* LOC: DCG 2006, V, page 198. Calc 17.4.2.2. *)
13245 (* I am not including this inequality because I don't see that it is needed.
13246 Ferguson gives a special boundary case of the inequality 9046001781 here, because
13247 he sees the dimension reduction as not applying in a boundary case. It seems
13248 to me that dimension reduction in the previous ineq is entirely justified. *)
13253 (* LOC: DCG 2006, V, page 198. Calc 17.4.2.3. *)
13254 (* LOC: DCG 2006, V, page 199. Calc 17.4.2.4. *)
13255 (* LOC: DCG 2006, V, page 199. Calc 17.4.2.5. *)
13256 (* LOC: DCG 2006, V, page 199. Calc 17.4.2.6. *)
13258 (* Ferguson separates the following two interval calculations into four cases,
13259 depending on things like derivative information,
13260 dimension reduction, a separate calculation in a small
13261 neighborhood of the tight corner at (2,2,2,2,2,Sqrt[8]), etc.
13262 I am combining them here. Ferguson's discussion may be needed in their formal
13265 CCC pp_b typo fixed.
13267 Bound: 0.118099592077
13269 Point: [4.00593290879, 4.00593290879, 4.000991016, 4.02090803522, 4.02090803742, 7.99999120025]
13276 [((#4.0),x1,square_2t0);
13277 ((#4.0),x2,square_2t0);
13278 ((#4.0),x3,square_2t0);
13279 ((#4.0),x4,square_2t0);
13280 ((#4.0),x5,square_2t0);
13281 (square_2t0,x6,(#8.0))
13283 ( (vor_analytic_x x1 x2 x3 x4 x5 x6 + pp_m * sol_x x1 x2 x3 x4 x5 x6 < pp_b / (#2.0)) \/
13284 (eta_x x1 x2 x6 < sqrt2) \/
13285 (vor_analytic_x x1 x2 x3 x4 x5 x6 < -- (#1.04)* pt))`;;
13290 CCC pp_b typo fixed.
13292 Bound: 0.119559508184
13294 Point: [4.00000394962, 4.00000394962, 4.00000197481, 4.0001220805, 4.0001220805, 7.99999999627]
13299 [((#4.0),x1,square_2t0);
13300 ((#4.0),x2,square_2t0);
13301 ((#4.0),x3,square_2t0);
13302 ((#4.0),x4,square_2t0);
13303 ((#4.0),x5,square_2t0);
13304 (square_2t0,x6,(#8.0))
13306 ( (vor_analytic_x x1 x2 x3 x4 x5 x6 + pp_m * sol_x x1 x2 x3 x4 x5 x6 < pp_b / (#2.0)) \/
13307 (eta_x x4 x5 x6 < sqrt2) \/
13308 (vor_analytic_x x1 x2 x3 x4 x5 x6 < -- (#1.04)* pt))`;;
13314 (* LOC: DCG 2006, V, page 199. Calc 17.4.3.1. *)
13315 (* upright quarters in an octahedron *)
13319 [(square_2t0,x1,(#8.0));
13320 (square (#2.2),x2,square_2t0);
13321 ((#4.0),x3,square_2t0);
13322 ((#4.0),x4,square_2t0);
13323 ((#4.0),x5,square_2t0);
13324 ((#4.0),x6,square_2t0)
13326 ( (octavor_analytic_x x1 x2 x3 x4 x5 x6 < -- (#0.52) * pt) \/
13327 (eta_x x1 x2 x6 < sqrt2))`;;
13332 [(square_2t0,x1,(#8.0));
13333 (square (#2.2),x2,square_2t0);
13334 ((#4.0),x3,square_2t0);
13335 ((#4.0),x4,square_2t0);
13336 ((#4.0),x5,square_2t0);
13337 ((#4.0),x6,square_2t0)
13339 ( (octavor_analytic_x x1 x2 x3 x4 x5 x6 < -- (#0.52) * pt) \/
13340 (eta_x x1 x3 x5 < sqrt2))`;;
13344 [(square_2t0,x1,(#8.0));
13345 (square (#2.2),x2,square_2t0);
13346 ((#4.0),x3,square_2t0);
13347 ((#4.0),x4,square_2t0);
13348 ((#4.0),x5,square_2t0);
13349 ((#4.0),x6,square_2t0)
13351 (gamma_x x1 x2 x3 x4 x5 x6 < -- (#0.52) * pt)`;;
13354 (* LOC: DCG 2006, V, page 199. Calc 17.4.3.2. *)
13358 [(square_2t0,x1,square (#2.716));
13359 ((#4.0),x2,square (#2.2));
13360 ((#4.0),x3,square (#2.2));
13361 ((#4.0),x4,square (#2.2));
13362 ((#4.0),x5,square (#2.2));
13363 ((#4.0),x6,square (#2.2))
13365 ((gamma_x x1 x2 x3 x4 x5 x6 + pp_c * dih_x x1 x2 x3 x4 x5 x6 < pp_d) \/
13366 (gamma_x x1 x2 x3 x4 x5 x6 < -- (#1.04) * pt))`;;
13368 (* LOC: DCG 2006, V, page 200. Calc 17.4.3.3. *)
13372 [(square (#2.716),x1,(#8.0));
13373 ((#4.0),x2,square (#2.2));
13374 ((#4.0),x3,square (#2.2));
13375 ((#4.0),x4,square_2t0);
13376 ((#4.0),x5,square (#2.2));
13377 ((#4.0),x6,square (#2.2))
13379 ((gamma_x x1 x2 x3 x4 x5 x6 + pp_m * sol_x x1 x2 x3 x4 x5 x6
13380 + (#0.14) * dih_x x1 x2 x3 x4 x5 x6 < pp_b / (#4.0) + (#0.14)* pi/ (#2.0)) \/
13381 (gamma_x x1 x2 x3 x4 x5 x6 < -- (#1.04) * pt) \/
13382 (eta_x x1 x2 x6 > sqrt2) \/
13383 (eta_x x1 x3 x5 > sqrt2))`;;
13386 (* LOC: DCG 2006, V, page 200. Calc 17.4.3.4. *)
13388 CCC Fixed typo. Sign on eta_x was reversed.
13391 Bound: 0.0249615271277
13393 Point: [7.89609717439, 4.000001105, 4.000001105, 6.30008811007, 4.00000159981, 4.00000159981]
13398 [(square (#2.716),x1,square (#2.81));
13399 ((#4.0),x2,square (#2.2));
13400 ((#4.0),x3,square (#2.2));
13401 ((#4.0),x4,square_2t0);
13402 ((#4.0),x5,square (#2.2));
13403 ((#4.0),x6,square (#2.2))
13405 ((octavor_analytic_x x1 x2 x3 x4 x5 x6 + pp_m * sol_x x1 x2 x3 x4 x5 x6
13406 + (#0.14) * dih_x x1 x2 x3 x4 x5 x6 < pp_b / (#4.0) + (#0.14)* pi/ (#2.0)) \/
13407 (octavor_analytic_x x1 x2 x3 x4 x5 x6 < -- (#1.04) * pt) \/
13408 (eta_x x1 x2 x6 < sqrt2))`;;
13410 (* LOC: DCG 2006, V, page 200. Calc 17.4.3.5. *)
13414 [(square (#2.81),x1,(#8.0));
13415 ((#4.0),x2,square (#2.2));
13416 ((#4.0),x3,square (#2.2));
13417 ((#4.0),x4,square_2t0);
13418 ((#4.0),x5,square (#2.2));
13419 ((#4.0),x6,square (#2.2))
13421 ((gamma_x x1 x2 x3 x4 x5 x6 + pp_m * sol_x x1 x2 x3 x4 x5 x6
13422 + (#0.054) * dih_x x1 x2 x3 x4 x5 x6 + (#0.00455) * (x1- (#8.0)) <
13423 pp_b / (#4.0) + (#0.054)* pi/ (#2.0)) \/
13424 (gamma_x x1 x2 x3 x4 x5 x6 < -- (#1.04) * pt) \/
13425 (eta_x x1 x2 x6 > sqrt2) \/
13426 (eta_x x1 x3 x5 > sqrt2))`;;
13429 (* LOC: DCG 2006, V, page 200. Calc 17.4.3.6. *)
13433 [(square (#2.81),x1,(#8.0));
13434 ((#4.0),x2,square (#2.2));
13435 ((#4.0),x3,square (#2.2));
13436 ((#4.0),x4,square_2t0);
13437 ((#4.0),x5,square (#2.2));
13438 ((#4.0),x6,square (#2.2))
13440 ((octavor_analytic_x x1 x2 x3 x4 x5 x6 + pp_m * sol_x x1 x2 x3 x4 x5 x6
13441 + (#0.054) * dih_x x1 x2 x3 x4 x5 x6 - (#0.00455) * (x1- (#8.0)) <
13442 pp_b / (#4.0) + (#0.054)* pi/ (#2.0)) \/
13443 (octavor_analytic_x x1 x2 x3 x4 x5 x6 < -- (#1.04) * pt) \/
13444 (eta_x x1 x2 x6 > sqrt2))`;;
13447 (* LOC: DCG 2006, V, page 201. Calc 17.4.4.1. *)
13448 (* pure Voronoi quad clusters, sigma is sqrt-2 truncated Voronoi *)
13453 [((#4.0),x1,square_2t0);
13454 ((#4.0),x2,square_2t0);
13455 ((#4.0),x3,square_2t0);
13456 ((#4.0),x4,square_2t0);
13457 ((#4.0),x5,square_2t0);
13458 (square (#2.84),x6,(#16.0))
13460 ((vort_x x1 x2 x3 x4 x5 x6 sqrt2 + pp_m * sol_x x1 x2 x3 x4 x5 x6
13461 < pp_b / (#2.0)) \/
13462 (sol_x x1 x2 x3 x4 x5 x6 < pp_solt0) \/
13464 (vort_x x1 x2 x3 x4 x5 x6 sqrt2 < -- (#1.04) * pt))`;;
13467 (* LOC: DCG 2006, V, page 201. Calc 17.4.4.... *)
13468 (* See note in DCG errata. We need to check that each half is nonpositive for the proof
13469 of Lemma DCG 16.7, page 182.
13477 (* LOC: DCG 2006, V, page 201. Calc 17.4.4.2. *)
13478 (* LOC: DCG 2006, V, page 201. Calc 17.4.4.3. *)
13479 (* pure Voronoi quad clusters, sigma is sqrt-2 truncated Voronoi *)
13481 (* This is separated into 2 cases in Ferguson. *)
13485 [((#4.0),x1,square_2t0);
13486 ((#4.0),x2,square_2t0);
13487 ((#4.0),x3,square_2t0);
13488 ((#4.0),x4,square_2t0);
13489 ((#4.0),x5,square_2t0);
13490 ((#8.0),x6,square (#2.84))
13492 ((vort_x x1 x2 x3 x4 x5 x6 sqrt2 + pp_m * sol_x x1 x2 x3 x4 x5 x6
13493 < pp_b / (#2.0)) \/
13494 (sol_x x1 x2 x3 x4 x5 x6 < pp_solt0) \/
13496 (vort_x x1 x2 x3 x4 x5 x6 sqrt2 < -- (#1.04) * pt))`;;
13499 (* LOC: DCG 2006, V, page 201. Calc 17.4.4.4. *)
13500 (* pure Voronoi quad clusters, sigma is sqrt-2 truncated Voronoi *)
13506 [((square ((#4.0)/(#2.51))),x,square_2t0);
13507 ((#8.0),ds,((#2.0)* square_2t0))
13509 ((vort_x (#4.0) (#4.0) (#4.0) x x ds sqrt2 + pp_m * sol_x (#4.0) (#4.0) (#4.0) x x ds
13510 < pp_b / (#2.0)) \/
13511 (vort_x (#4.0) (#4.0) (#4.0) x x ds sqrt2 < -- (#0.52) * pt) \/
13512 (sol_x (#4.0) (#4.0) (#4.0) x x ds < pp_solt0) \/
13513 ((#2.0) * x < ds))`;;
13515 (* LOC: DCG 2006, V, page 201. Calc 17.4.4.5. *)
13516 (* pure Voronoi quad clusters, sigma is sqrt-2 truncated Voronoi *)
13522 [((square ((#4.0)/(#2.51))),x1,square_2t0);
13523 ((square ((#4.0)/(#2.51))),x2,square_2t0)
13525 ((vort_x (#4.0) (#4.0) (#4.0) x1 x1 (#8.0) sqrt2 + vort_x (#4.0) (#4.0) (#4.0) x2 x2 (#8.0) sqrt2
13526 + pp_m * (sol_x (#4.0) (#4.0) (#4.0) x1 x1 (#8.0) + sol_x (#4.0) (#4.0) (#4.0) x2 x2 (#8.0))
13528 (vort_x (#4.0) (#4.0) (#4.0) x1 x1 (#8.0) sqrt2 + vort_x (#4.0) (#4.0) (#4.0) x2 x2 (#8.0) sqrt2 < -- (#1.04) * pt) \/
13529 (sol_x (#4.0) (#4.0) (#4.0) x1 x1 (#8.0) + sol_x (#4.0) (#4.0) (#4.0) x2 x2 (#8.0) < (#2.0) * pp_solt0))`;;
13532 (* LOC: DCG 2006, V, page 201. Calc 17.4.5.1. DCG, V, page 174, Theorem 16.1. *)
13533 (* This 91-term polynomial is used to justify dimension reduction for vol_analytic_x. *)
13534 (* Ferguson states two cases, but the second case covers the first as well. *)
13536 (* This has been formally verified by R. Zumkeller in COQ on March 6 2008.
13539 My tactic reports -451149333733932001/156250000000000 (approximately
13540 -2887.36) as the sharp maximum of the left-hand side. Mathematica
13541 seems to agree. As you can see below conversion to the Bernstein basis
13542 was sufficient, no subdivisions are needed.
13544 Time Eval vm_compute in min_bb_Q_Ff steps (prec (-10))
13545 (ply_mgm.mdlN_of_rngN (-ferguson)).
13546 = (451149333733932001 # 156250000000000,
13547 451149333733932001 # 156250000000000, true,
13548 (0%nat, 0%nat, 0%nat))
13549 : Q * Q * bool * (nat * nat * nat)
13550 Finished transaction in 2. secs (1.963554u,0.021168s)
13552 (* upper bound on x4 changed 3/7/08, new domain *)
13555 (* moved 2298281931 to inequality_spec.ml *)
13559 (* End of Sphere Packings V, DCG, Ferguson's thesis *)