(* ========================================================================== *) (* FLYSPECK - BOOK FORMALIZATION *) (* *) (* Chapter: Nonlinear *) (* Author: Thomas C. Hales *) (* Date: 2010-05-30 *) (* ========================================================================== *) (* File of the nonlinear inequalities to be verified by interval arithmetic. *) flyspeck_needs "general/sphere.hl";; flyspeck_needs "nonlinear/ineqdata3q1h.hl" ;; (* XX some Flypaper[] tags exist, without text attribution *) module Ineq = struct open Sphere;; (* The nonlinear inequality data has become too widely dispersed over the project directories. This file is meant to be the authoritative central repository for inequalities. Old sources of information: (svn 1678 2010-04-26 of NonlinearInequalities.wiki) software_guide.tex (svn 1760) *) let ineqdoc = ref [];; let addtex (a,b,c) = (ineqdoc := (a,b,c)::!ineqdoc);; let ineqs = ref ([]:ineq_datum list);; let add i = ( ineqs:= i :: !ineqs; if (List.mem Tex i.tags) then addtex (Ineqdoc,i.idv,i.doc) else () );; let skip _ = ();; let remove idv = ( ineqs := filter (fun t -> not(t.idv=idv)) (!ineqs) );; let getexact idv = filter (fun t -> (t.idv = idv)) (!ineqs);; let getprefix idv = filter (fun t -> (String.length idv <= String.length t.idv) && (String.sub t.idv 0 (String.length idv) = idv)) (!ineqs);; let getfield r = filter (fun t -> mem r t.tags) (!ineqs);; (* special tags have all been eliminated. These fields can all be deprecated. let rec has_delta = function | [] -> false | Delta126min t :: _ -> true | Delta126max t :: _ -> true | Delta135min t :: _ -> true | Delta135max t :: _ -> true | Set_rad2 :: _ -> true | _ :: a -> has_delta a;; let has_special_tags() = filter (function t -> has_delta t.tags) (!Ineq.ineqs);; *) let flypaper_ids idv = List.flatten( map (function Flypaper s -> s | _ -> []) idv.tags);; let all_flypaper() = unions (map flypaper_ids (!ineqs));; let do_betas x = all_forall (snd(dest_eq(concl(BETAS_CONV x))));; let mk_tplate template r = do_betas(list_mk_comb (template, r));; (************** DATA SECTION ******************************) addtex (Section,"Packing -- Marchal Inequality","");; addtex(Comment,"", "This first series shows that a Marchal 4-cell without any critical edges is non-negative. ");; (******************************************************************************) (* MARCHAL 4-CELL WITH NO CRITICAL EDGES NONNEGATIVITY *) (******************************************************************************) let TSKAJXY_DERIVED = { idv= "TSKAJXY-DERIVED"; ineq = (all_forall `ineq [ (#2.0,y1,sqrt8); (#2.0,y2,sqrt8); (#2.0,y3,sqrt8); (#2.0,y4,sqrt8); (#2.0,y5,sqrt8); (#2.0,y6,sqrt8) ] (( ~(critical_edge_y y1) /\ ~(critical_edge_y y2) /\ ~(critical_edge_y y3) /\ ~(critical_edge_y y4) /\ ~(critical_edge_y y5) /\ ~(critical_edge_y y6) /\ &0 < delta_y y1 y2 y3 y4 y5 y6 /\ rad2_y y1 y2 y3 y4 y5 y6 < &2) ==> (gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun >= &0) )`); doc="If a four-cell does not have any critical edges then it is non-negative. Critical edge, delta,rad conditions added 2012-06. A derivation of this from other inequalities appears in merge_ineq.hl. "; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Cfsqp;Xconvert;Derived [];Branching;Eps 1.0e-12;Sharp]; };; (* The following make up merge_ineq.tsk_hyp *) add { idv= "TSKAJXY-TADIAMB"; ineq = (all_forall `ineq [ (&2 * hplus,y1,sqrt8); (&2 * hplus,y2,sqrt8); (#2.0,y3,sqrt8); (#2.0,y4,sqrt8); (#2.0,y5,sqrt8); (#2.0,y6,sqrt8) ] ( (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2))`); doc= "This is a reduction step for main modified Marchal inequality (for lmfun). Cannot have two adjacent long edges."; tags = [Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Cfsqp;Xconvert]; };; skip { idv= "TSKAJXY-RIBCYXU"; ineq = (all_forall `ineq [ (#2.0,y1,&2 * hminus); (#2.0,y2,&2 * hminus); (#2.0,y3,&2 * hminus); (#2.0,y4,&2 * hminus); (#2.0,y5,&2 * hminus); (#2.0,y6,&2 * hminus) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun >= &0) )`); doc= "If a simplex does not have any critical edges then it is non-negative. This had long run times, so it has been replaced with the two subcases -sharp and -sym that do symmetry reductions of the domain."; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Cfsqp;Xconvert;Branching;Eps 1.0e-12;Sharp;Deprecated]; };; add { idv= "TSKAJXY-RIBCYXU sharp"; ineq = (all_forall `ineq [ (#2.0,y1,#2.001); (#2.0,y2,#2.001); (#2.0,y3,#2.001); (#2.0,y4,#2.001); (#2.0,y5,#2.001); (#2.0,y6,#2.001) ] (gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun >= &0)`); doc= "If a simplex does not have any critical edges then it is non-negative. "; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Cfsqp;Xconvert;Branching;Eps 1.0e-12;Sharp]; };; add { idv= "TSKAJXY-RIBCYXU sym"; ineq = (all_forall `ineq [ (#2.001,y1,&2 * hminus); (#2.0,y2,&2 * hminus); (#2.0,y3,&2 * hminus); (#2.0,y4,&2 * hminus); (#2.0,y5,&2 * hminus); (#2.0,y6,&2 * hminus) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun > &0) \/ (y1 < y2) \/ (y1 < y3) \/ (y1 < y4) \/ (y1 < y5) \/ (y1 < y6) \/ (y2 < y3) \/ (y2 < y5) \/ (y2 < y6))`); doc= " If a simplex does not have any critical edges then it is non-negative. By symmetry, wlog y1 is largest and y2 is largest among y2,y3,y5,y6"; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Cfsqp;Xconvert;Branching]; };; skip { idv= "TSKAJXY-IYOUOBF"; ineq = (all_forall `ineq [ (&2 * hplus,y1,sqrt8); (#2.0,y2,&2 * hminus); (#2.0,y3,&2 * hminus); (#2.0,y4,&2 * hminus); (#2.0,y5,&2 * hminus); (#2.0,y6,&2 * hminus) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun >= &0) )`); doc= "If a simplex does not have any critical edges then it is non-negative. Replaced May 24, 2011 with the symmetrized version to improve execution time."; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Cfsqp; Xconvert;Branching;Eps 1.0e-12;Sharp;Deprecated]; };; add { idv= "TSKAJXY-IYOUOBF sym"; ineq = (all_forall `ineq [ (&2 * hplus,y1,sqrt8); (#2.001,y2,&2 * hminus); (#2.0,y3,&2 * hminus); (#2.0,y4,&2 * hminus); (#2.0,y5,&2 * hminus); (#2.0,y6,&2 * hminus) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun >= &0) \/ (y2 < y3 ) \/ (y2 < y5) \/ (y2 < y6) )`); doc= "If a simplex does not have any critical edges then it is non-negative. By symmetry, we may assume that y2 is the longest of y2,y3,y5,y6"; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Cfsqp;Xconvert;Branching;Eps 1.0e-12;Sharp]; };; add { idv= "TSKAJXY-IYOUOBF sharp v2"; ineq = (all_forall `ineq [ (&2 * hplus,y1,sqrt8); (#2.0,y2,#2.001); (#2.0,y3,#2.001); (#2.0,y4,&2 * hminus); (#2.0,y5,#2.001); (#2.0,y6,#2.001) ] (gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun >= &0)`); doc= "If a simplex does not have any critical edges then it is non-negative. Upper bound on y4 corrected from 2.001 to 2 * hminus on 2012-06-15"; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Cfsqp;Xconvert;Branching;Eps 1.0e-12;Sharp]; };; skip { idv= "TSKAJXY-WKGUESB"; ineq = (all_forall `ineq [ (&2 * hplus,y1,sqrt8); (#2.01,y2,&2 * hminus); (#2.0,y3,&2 * hminus); (&2 * hplus,y4,sqrt8); (#2.0,y5,&2 * hminus); (#2.0,y6,&2 * hminus) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun > &0) )`); doc="If a simplex does not have any critical edges then it is non-negative. Case opposite edges y1,y4 long, and some other edge (y2) at least 2.01. May 24, 2011. This was replaced with 'TSKAJXY-WKGUESB sym' that reduces the domain using symmetry, because this was a slow calculation. "; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Branching;Cfsqp;Xconvert;Deprecated]; };; add { idv= "TSKAJXY-WKGUESB sym"; ineq = (all_forall `ineq [ (&2 * hplus,y1,sqrt8); (#2.01,y2,&2 * hminus); (#2.0,y3,&2 * hminus); (&2 * hplus,y4,sqrt8); (#2.0,y5,&2 * hminus); (#2.0,y6,&2 * hminus) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun > &0) \/ (y2 < y3) \/ (y2 < y5) \/ (y2 < y6) \/ (y1 < y4) )`); doc="If a simplex does not have any critical edges then it is non-negative. Add symmetry reductions. Case y1,y4 long and longest of the other edges is at least 2.01. Wlog, by symmetry y2. Wlog, by tetrahedral symmetry, y1 longer than y4. "; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Branching;Cfsqp;Xconvert]; };; add { idv= "TSKAJXY-XLLIPLS"; ineq = (all_forall `ineq [ (&2 * hplus,y1,sqrt8); (#2.0,y2,#2.01); (#2.0,y3,#2.01); (&2 * hplus,y4,#2.8); (#2.0,y5,#2.01); (#2.0,y6,#2.01) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun > &0) )`); doc= "If a simplex does not have any critical edges then it is non-negative. Case y1,y4 long, y1 <= 2.8, and all other edges at most 2.01. Swapped upper bounds on y1,y4 on 2012-06-15. This form is equivalent, but easier to use. "; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Branching;Cfsqp;Xconvert;]; };; add { idv= "TSKAJXY-eulerA"; ineq = (all_forall `ineq [ (#2.8 pow 2,x1,&8); (&4,x2,#2.01 pow 2); (&4,x3,#2.01 pow 2); (#2.8 pow 2,x4,&8); (&4,x5,#2.01 pow 2); (&4,x6,#2.01 pow 2) ] (&0 < eulerA_x x1 x2 x3 x4 x5 x6) `); doc="Easy bound needed for 'TSKAJXY-GXSABWC DIV'. Added 2012-06."; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Cfsqp;Derived []]; };; add { idv= "TSKAJXY-delta_x4"; ineq = (all_forall `ineq [ (&4,x1,#2.01 pow 2); (&4,x2,#2.01 pow 2); (#2.8 pow 2,x3,&8); (&4,x4,#2.01 pow 2); (&4,x5,#2.01 pow 2); (#2.8 pow 2,x6,&8) ] (&0 < delta_x4 x1 x2 x3 x4 x5 x6) `); doc="Easy bound needed for 'TSKAJXY-GXSABWC DIV'. Added 2012-06."; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Cfsqp;Derived []]; };; add { idv= "TSKAJXY-GXSABWC DIV"; ineq = (all_forall `ineq [ (#2.8 pow 2,x1,&8); (&4,x2,#2.01 pow 2); (&4,x3,#2.01 pow 2); (#2.8 pow 2,x4,&8); (&4,x5,#2.01 pow 2); (&4,x6,#2.01 pow 2) ] ((&1 / &12 - ( (&2 * mm1 / pi) * (sol_euler_x_div_sqrtdelta x1 x2 x3 x4 x5 x6 + sol_euler345_x_div_sqrtdelta x1 x2 x3 x4 x5 x6 + sol_euler156_x_div_sqrtdelta x1 x2 x3 x4 x5 x6 + sol_euler246_x_div_sqrtdelta x1 x2 x3 x4 x5 x6) - (&8 * mm2 / pi) * ( ldih2_x_div_sqrtdelta_posbranch x1 x2 x3 x4 x5 x6 + ldih3_x_div_sqrtdelta_posbranch x1 x2 x3 x4 x5 x6 + ldih5_x_div_sqrtdelta_posbranch x1 x2 x3 x4 x5 x6 + ldih6_x_div_sqrtdelta_posbranch x1 x2 x3 x4 x5 x6 ) ) >= &0) \/ (delta_x x1 x2 x3 x4 x5 x6 < &0) )`); doc= "If a simplex does not have any critical edges then it is non-negative. This is a degenerate calculation, because the volume of the simplex tends to zero. Case y1, y4 long, and all variables near the critical point."; tags=[Marchal;Flypaper["OXLZLEZ";"TSKAJXY";];Tex;Cfsqp;]; };; (******************************************************************************) (* MARCHAL CELL_CLUSTER 4-CELLS, AT LEAST 5 LEAVES, ZTG4 SERIES. *) (******************************************************************************) addtex(Section,"Packing Cluster Inequality -- Five or more leaves","");; addtex(Comment,"", " % cc:5bl Let $B$ be the set of cells in the cluster that lie between any two consecutive leaves. $B$ is either a singleton set containing a $4$-cell, or a set of three cells: a $2$-cell and two adjacent $3$-cells. Write $\\op{azim}(B)$ for the azimuth angle formed by the two leaves. Let \\[ a= 0.0560305, \\textand b= -0.0445813. \\] The constants $a$ and $b$ must satisfy $5 a + b (2\\pi) >0$, for the $5$-leaf case of the cluster inequality. The circumradius of a triangle with sides $h_-,h_-,h_-$ is greater than $\\sqrt2$, so in $4$-cells some edge next to the spine is subcritical. Without loss of generality, we can assume it is $y_2$. We have the inequality for all $4$-cells: \\[ \\gamma(X,L) \\op{wt}(X) + \\beta(\\e,X) \\ge a + b \\azim(X). \\] This has many cases \\ineq{ZTGIJCF4 i3 i4 i5 i6}, depending on which edges $y_3,\\ldots,y_6$ are subcritical, critical, or supercritical. \\begin{itemize}\\wasitemize \\item \\case{1821661595} A $4$-cell $X$ along a spine $e$ satisfies \\[ \\gamma_L(X)\\op{wt}(X) + \\beta(e,X) \\ge a + b\\,\\op{azim}(X), \\] \\item \\case{7907792228} The $2$-cell $X_2$ and two $3$-cells $X_1,X_3$ that flank it along a spine $e$ satisfy \\[ \\sum_{i=1}^3 \\left(\\gamma_L(X_i)\\op{wt}(X_i) + \\beta(e,X_i)\\right)\\ge a + b\\,\\sum_{i=1}^3\\op{azim}(X_i). \\] \\end{itemize}\\wasitemize Then \\[ \\sum_{X\\in B} \\gamma(X,L)\\op{wt}(X) + \\beta(\\ee,X) \\ge a + b\\,\\op{azim}(B). \\] It follows that \\[ \\Gamma(Z) \\ge 5 a + b\\, (2\\pi) > 0. \\] ");; add { idv = "ZTGIJCF0"; ineq = all_forall `ineq [(&1,dummy,&1)] ( &5 * a_spine5 + b_spine5 * &2 * pi > &0)`; doc = "Calculation of constants for 5 or more leaves"; tags=[Marchal;Flypaper["OXLZLEZ";];Tex;Cfsqp]; };; (* F4 *) let template_F4 = `\ y3m y3M y4m y4M y5m y5M y6m y6M w m. ineq [(&2 * hminus, y1, &2 * hplus); (&2 ,y2, &2 *hminus); (y3m,y3,y3M); (y4m,y4,y4M); (y5m,y5,y5M); (y6m,y6,y6M)] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun / &w + &m *beta_bump_lb > a_spine5 + b_spine5 * dih_y y1 y2 y3 y4 y5 y6) \/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2))`;; let mk_ineq_F4 i3 i4 i5 i6 = let x i = List.nth [`&2`; `&2 * hminus`; `sqrt8`] i in let X i = x (i+1) in let mid i = if (i=1) then 1 else 0 in let w = 1 + mid i3 + mid i4 + mid i5 + mid i6 in let m = if (w =2) && (i4 = 1) then `1` else `0` in mk_tplate template_F4 [x i3;X i3; x i4;X i4; x i5;X i5; x i6 ;X i6; mk_small_numeral w; m];; (* bug fixed July 15, 2011. To many cases were being generated. 0 <, changed to 1= *) let split_F4 i3 i4 i5 i6 = let split i = (1 = List.nth [1;0;i3;i4;i5;i6] i) in filter split [0;1;2;3;4;5];; (* rewrote May 25, 2011 let split_F4 i3 i4 i5 i6 = let sp ls i pos = if (i=1) then ls @ [pos] else ls in let l0 = [0] in let l3 = sp l0 i3 2 in let l4 = sp l3 i4 3 in let l5 = sp l4 i5 4 in sp l5 i6 5;; *) let make_F4 i3 i4 i5 i6 = { idv = Printf.sprintf "ZTGIJCF4 %d %d %d %d 1821661595" i3 i4 i5 i6; ineq = mk_ineq_F4 i3 i4 i5 i6; doc = "This is the $4$-cell inequality for five or more leaves."; tags=(if (i3,i4,i5,i6)=(0,0,0,0) then [Tex] else [] ) @ [Marchal;Cfsqp;Flypaper["OXLZLEZ";];Xconvert;Penalty(50.0,500.0);Branching;Split (split_F4 i3 i4 i5 i6)]; };; for i3=0 to 1 do for i4 = 0 to 1 do for i5 = 0 to 1 do for i6 = 0 to 1 do add(make_F4 i3 i4 i5 i6) done done done done;; add { idv = "MKFKQWU"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus ); (&2*hminus ,y2,sqrt8 ); (&2*hminus,y3,sqrt8); (&2,y4, sqrt8); (&2*hminus,y5,sqrt8 ); (&2*hminus,y6,sqrt8 ) ] (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2)`; doc= "This is a reduction step for 5-leaf ineq on 4-cells, with critical edge $y_1$. It justifies that the shortest edge $y_2$ adjacent to the spine must be at most 2hminus in 'ZTGIJCF4'. Added 2012-06. "; tags = [Flypaper["OXLZLEZ"];Tex;Cfsqp;Xconvert]; };; add { idv = "MKFKQWU halfwt"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus ); (&2*hplus ,y2,sqrt8 ); (&2,y3,sqrt8); (&2*hminus,y4, sqrt8); (&2,y5,sqrt8 ); (&2,y6,sqrt8 ) ] (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2)`; doc= "This is a reduction step for 5-leaf ineq on 4-cells, with critical edge $y_1$. It means that a 4-cell with a beta bump, has edges y2,y3,y5,y6 subcritical. Used to justify beta approximations in 'ZTGIJCF4'. Added 2012-06. "; tags = [Flypaper["OXLZLEZ"];Tex;Cfsqp;Xconvert]; };; (******************************************************************************) (* MARCHAL 4-CELL QX LOWER BOUNDS *) (******************************************************************************) addtex(Comment,"", " Call a $4$-cell a quarter, when it has exactly one critical edge and all other edges of the simplex have length at most $2 h_-$. (Actually, strictly less than 2h_-, because if = 2h_-, then critical.) The weight of any quarter is $1$. Here we treat 4-cells with at least one critical edge, but not a quarter. ");; add { idv = "GLFVCVK4 2477216213"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2,y2,sqrt8); (&2,y3,sqrt8); (&2,y4,sqrt8); (&2,y5,sqrt8); (&2,y6,sqrt8) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun > &0)\/ (norm2hh y1 y2 y3 y4 y5 y6 < (hplus- hminus) pow 2) \/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2))`; (* norm2hh was <= *) doc = " OXLZLEZ.mod 'gamma_qx' \\claim{If $X$ is a $4$-cell that is not a quarter, then $\\gamma(X,L)\\ge0$.} Indeed, if no edge of $X$ is critical, we use 'TSKAJXY-DERIVED'. If some edge of $X$ is critical, then we may label the edges so that it is the first, and this computer calculation treats it. If the norm condition holds, then it is a quarter. This norm condition seems easier to check in practice than the conditions defining a quarter. "; tags=[Marchal;Flypaper["OXLZLEZ";];Tex;Xconvert;Branching;Split[0;1;2;3;4;5]]; };; add { idv = "GLFVCVK4a 8328676778"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus ); (&2 ,y2,&2 * hminus ); (&2,y3,&2 * hminus); (&2 * hminus ,y4,&2 * hplus ); (&2,y5,&2 * hminus ); (&2,y6,&2 * hminus ) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun / &2 + beta_bump_force_y y1 y2 y3 y4 y5 y6 > &0)\/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2))`; doc = " OXLZLEZ.mod 'gamma_qx' Let $\\gamma$ be given by Definition~\\ref{def:gammaL}, $\\op{wt}$ by Definition~\\ref{def:wt}, and $\\beta$ by Definition~\\ref{def:beta}. If $X$ is any $k$-cell with $k\\in\\{2,3,4\\}$, and if $X$ is not a quarter, then then \\[ \\gamma(X,L) \\op{wt}(X) + \\beta(\\ee,X)\\ge 0. \\] % gammaL is nonneg on quarters. cc:qtr In fact, $\\beta(\\ee,X)=0$, except possibly when $X$ is a $4$-cell with oppositely arranged critical edges. Hence in most cases, it is enough to check the simpler inequality $\\gamma(X,L)\\ge0$. This is the verification of the case where the factor $\\beta$ matters and where all edges are critical or subcritical. If some edge is supercritical, then the circumradius is at least \\[ \\op{rad}(2h_-,2,2,2h_-,2h_+,2) > \\sqrt2, \\] and the simplex is not a $4$-cell. "; tags=[Marchal;Flypaper["OXLZLEZ";];Tex;Cfsqp;Xconvert;Branching;Split[0;3]]; };; add { idv = "GLFVCVK4 2477216213 y4crit"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2 * hminus,y2,&2 * hplus ); (&2,y3,&2 * hminus); (&2 * hminus,y4,&2 * hplus); (&2,y5,&2 * hminus); (&2,y6,&2 * hminus) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun / &3 > #0.0057)\/ (eta_y y1 y2 y6 pow 2 < #1.34 pow 2) \/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2))`; doc = " OXLZLEZ.mod 'gamma_qx' QX Non-quarter, non beta-half, eta126>1.34, y3 y5 subcritical ==> gwt > 0.0057. Case: y4 critical Non-beta-half because y2 (adj to spine) is not subcritical. y2 upper bound: Rad[2hminus,2hplus,2,2hminus,2,2]^2 > 2. y6 upper bound comes eta[2hminus,2hminus,2hminus]>sqrt2. "; tags=[Marchal;Flypaper["OXLZLEZ";"Jun2012"];Tex;Xconvert;Branching;Split[0;1;3]]; };; add { idv = "GLFVCVK4 2477216213 y4supercrit"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2,y2,&2 * h0); (&2,y3,&2 * hminus); (&2 * hplus,y4,sqrt8); (&2,y5,&2 * hminus); (&2,y6,&2 * hminus) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun / &2 > #0.0057)\/ (eta_y y1 y2 y6 pow 2 < #1.34 pow 2) \/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2))`; doc = " OXLZLEZ.mod 'gamma_qx' QX Non-quarter, non beta-half, eta126>1.34, y3 y5 subcritical ==> gwt > 0.0057. Case: y4 critical Non-beta-half nonquarter because y4 supercritical. y2 upper bound: Rad[2hminus,2h0,2,2hplus,2,2]^2 > 2. y6 upper bound comes eta[2hminus,2hminus,2hminus]>sqrt2. and wlog y6 < y2. "; tags=[Marchal;Flypaper["OXLZLEZ";"Jun2012"];Tex;Xconvert;Branching;Split[0]]; };; add { idv = "GLFVCVK4 2477216213 y4subcrit"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2 * hminus,y2,sqrt8); (&2,y3,&2 * hminus); (&2,y4,&2 * hminus); (&2,y5,&2 * hminus); (&2,y6,&2 * hminus) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun / &2 > #0.0057)\/ (eta_y y1 y2 y6 pow 2 < #1.34 pow 2) \/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2))`; doc = " OXLZLEZ.mod 'gamma_qx' QX Non-quarter, non beta-half, eta126>1.34, y3 y5 subcritical ==> gwt > 0.0057. Case: y4 subcritical Non-quarter because y2 (adj to spine) is not subcritical. y6 upper bound comes eta[2hminus,2hminus,2hminus]>sqrt2. "; tags=[Marchal;Flypaper["OXLZLEZ";"Jun2012"];Tex;Xconvert;Branching;Split[0;1]]; };; add { idv = "BIXPCGW 6652007036 a2"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2,y2, sqrt8); (&2,y3, sqrt8); (&2,y4, sqrt8); (&2,y5, sqrt8); (&2,y6, sqrt8) ] ((dih_y y1 y2 y3 y4 y5 y6 < #2.8) )`; doc = " OXLZLEZ.mod 'azim_c4' QX and QU If $X$ is a $4$-cell then $\\dih(X) < 2.8$."; tags=[Marchal;Flypaper["OXLZLEZ";];Cfsqp;Clusterlp;Tex;Xconvert]; };; add { idv = "BIXPCGW 7080972881 a2"; ineq = all_forall `ineq [(&2 * hminus,y1, &2 * hplus); (&2 * hminus,y2, sqrt8); (&2,y3, sqrt8); (&2,y4, sqrt8); (&2,y5, sqrt8); (&2,y6, sqrt8) ] ((dih_y y1 y2 y3 y4 y5 y6 < #2.3) )`; doc = " OXLZLEZ.mod 'g_qxd' QXD If $X$ is a $4$-cell with a critical edge next to the spine, then $\\dih(X) < 2.3$."; tags=[Marchal;Flypaper["OXLZLEZ";];Cfsqp;Clusterlp;Tex;Xconvert]; };; add { idv = "BIXPCGW 1738910218 a2"; ineq = all_forall `ineq [(&2 * hminus,y1, &2 * hplus); (&2,y2, sqrt8); (&2,y3, sqrt8); (&2,y4, &2 * hplus); (&2,y5, sqrt8); (&2,y6, sqrt8) ] ( (dih_y y1 y2 y3 y4 y5 y6 < #2.3) )`; doc = " OXLZLEZ.mod 'g_qxd' QXD If $X$ is a $4$-cell with a critical edge opposite spine, then $\\dih(X) < 2.3$."; tags=[Marchal;Flypaper["OXLZLEZ";];Cfsqp;Clusterlp;Tex;Xconvert]; };; add { idv = "BIXPCGW b"; ineq = all_forall `ineq [(&2 * hminus,y1, &2 * hplus); (&2,y2, sqrt8); (&2,y3, sqrt8); (&2,y4, &2 * hplus); (&2,y5, sqrt8); (&2,y6, sqrt8) ] ( (delta4_y y1 y2 y3 y4 y5 y6 > &0) \/ (delta_y y1 y2 y3 y4 y5 y6 > &60) \/ (delta_y y1 y2 y3 y4 y5 y6 < &0))`; doc = " NONQXD If $X$ is a $4$-cell with a critical edge opposite spine, then $\\dih(X) < 2.3$."; tags=[Marchal;Flypaper["OXLZLEZ";];Cfsqp;Clusterlp;Penalty (500.0,500.0);Tex;Xconvert]; };; add { idv = "BIXPCGW 7274157868 a"; ineq = all_forall `ineq [(&2 * hminus,y1, &2 * hplus); (&2,y2, &2 * hminus); (&2,y3, &2 * hminus); (&2 * hplus,y4, sqrt8); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun > #0.0057) \/ (dih_y y1 y2 y3 y4 y5 y6 < #2.3))`; doc = " OXLZLEZ.mod 'g_qxd' QXD If $X$ is a $4$-cell with a single critical edge (the spine), and if $\\dih(X)\\ge 2.3$, then $\\gamma(X,L) > 0.0057$. Domain restricted, Dec 1, 2012. Outside this, we have dih_y < 2.3 anyway."; tags=[Marchal;Flypaper["OXLZLEZ";];Cfsqp;Clusterlp;Tex;Xconvert;Branching;Split[0]]; };; add { idv = "QITNPEA 9939613598"; ineq = all_forall `ineq [(&2 * hminus,y1, &2 * hplus); (&2,y2, &2 * hminus); (&2,y3, &2 * hminus); (&2 * hplus,y4, sqrt8); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun - #0.00457511 - #0.00609451*dih_y y1 y2 y3 y4 y5 y6 > #0.0) \/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2))`; doc = " OXLZLEZ.mod 'azim2' FULLWT This is an inequality for nonquarter $4$-cells of weight $1$ used to prove the cluster inequality."; tags=[Marchal;Cfsqp;Flypaper["OXLZLEZ"];Clusterlp;Tex;Xconvert;Branching;Split[0]]; };; (******************************************************************************) (* MARCHAL CELL_CLUSTER 4-CELL INEQS QX (continued gaz7, gamma8, gaz9) *) (******************************************************************************) (* beta_bump_lb -> beta_bumpA_y 2010-06-23, false otherwise on (i3,i4,i5,i6) = (0,1,0,0) beta_bumpA_y -> beta_bump_force_y 2010-09-23 *) let mk_QITNPEA1 i3 i4 = let template = `\ y3m y3M y4m y4M w . ineq [(&2 * hminus, y1, &2 * hplus); (&2 ,y2, &2 *hminus); (y3m,y3,y3M); (y4m,y4,y4M); (&2 ,y5,&2 *hminus); (&2 ,y6,&2 * hminus)] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun / &w // + &m *beta_bump_lb > #0.0057) \/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2))` in let x i = List.nth [`&2`; `&2 * hminus`; `&2 * hplus`; `sqrt8`] i in let X i = x (i+1) in let mid i = if (i=1) then 1 else 0 in let w = 1 + mid i3 + mid i4 in (* let m = if (w =2) && (i4 = 1) then `1` else `0` in *) mk_tplate template [x i3;X i3; x i4;X i4; mk_small_numeral w; ];; let add_QITNPEA1 i3 i4 = (* if (i3+i4 = 0) then () else *) add{ idv = Printf.sprintf "QITNPEA1 %d %d 9063653052 A" i3 i4 ; ineq = mk_QITNPEA1 i3 i4 ; doc = " OXLZLEZ.mod 'gamma8' This is a $4$-cell (nonquarter) inequality. The four cell is assumed to have exactly one face along the spine with 2 subcritical (y2,y6). Note eta[2hminus,2hminus,2hminus]^2 > 2, so each face along the spine has at least one subcritical, this gives y5 subcritical, wlog. Dec 1, 2012. Removed beta_bump_lb, since y3 is never subcritical, using beta_bumpA_y. "; tags=(if (i3+i4=1) then [Tex] else []) @ [Marchal;Cfsqp;Flypaper["OXLZLEZ";];Xconvert;Penalty(50.0,500.0);Branching;Split (split_F4 i3 i4 0 0)]; };; for i3=1 to 2 do (* start at 1 to force the triangle (y1,y3,y5) not to be small *) for i4 = 0 to 2 do add_QITNPEA1 i3 i4 done done;; add{ idv = "QITNPEA 2134082733" ; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2 ,y2, &2 *hminus); (&2 ,y3,&2 * hminus); (&2 * hminus,y4,sqrt8); (&2 ,y5,&2 * hminus); (&2 ,y6,&2 * hminus)] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun / &2 + beta_bump_lb - #0.213849 + #0.119482*dih_y y1 y2 y3 y4 y5 y6 > #0.0 ) \/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2))`; doc = " OXLZLEZ.mod 'gaz9' This is a $4$-cell (nonquarter) inequality. The two edges along the spine are small."; tags=[Marchal;Cfsqp;Flypaper["OXLZLEZ";];Clusterlp;Xconvert;Tex;Branching;Split[0;3]]; };; (******************************************************************************) (* MARCHAL 4-CELL QU LOWER BOUNDS *) (******************************************************************************) add { idv = "FHBVYXZv2 a"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus ); (&2 ,y2,&2 * hminus ); (&2,y3,&2 * hminus); (&2,y4,&2 * hminus); (&2,y5,&2 * hminus ); (&2,y6,&2 * hminus ) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun > #0.0057)\/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2) \/ (eta_y y1 y3 y5 pow 2 < #1.34 pow 2))`; doc = " If $X$ is any quarter, with eta >= 1.34, then \\[ \\gamma(X,L) \\ge 0.0057. \\] Nov2012: Needed? Yes, see June 23, 2012 notes. "; tags=[Marchal;Flypaper["OXLZLEZ";"Jun2012"];Tex;Cfsqp;Xconvert;Branching;Split[0]]; };; add { idv = "FHBVYXZ a"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus ); (&2 ,y2,&2 * hminus ); (&2,y3,&2 * hminus); (&2,y4,&2 * hminus); (&2,y5,&2 * hminus ); (&2,y6,&2 * hminus ) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun > &0)\/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2) \/ (eta_y y1 y2 y6 pow 2 < #1.34 pow 2))`; doc = " OXLZLEZ.mod 'g_quqya' 'g_quqyb' %old idv: 1118115412, cc:2bl If $X$ is any quarter, and $Y$ is a $3$-cell that flanks it, then \\[ \\gamma(X,L) + \\gamma(Y,L) \\ge 0. \\] Nov2012, changed eta_y y1 y3 y5 to eta_y y1 y2 y6. "; (* + &0 * gamma3f y1 y3 y5 sqrt2 lmfun dropped *) tags=[Marchal;Flypaper["OXLZLEZ";];Tex;Cfsqp;Xconvert;Branching;Split[0]]; };; add { idv = "FHBVYXZ b"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus ); (&2,y2,&2 * hminus ); (&2,y3,&2 * hminus); (&2,y4,&2 * hminus); (&2,y5,&2 * hminus ); (&2,y6,&2 * hminus ) ] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun + gamma3f y1 y2 y6 sqrt2 lmfun > &0)\/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2) \/ (eta_y y1 y2 y6 pow 2 > #1.34 pow 2))`; doc = " OXLZLEZ.mod 'g_quqya' 'g_quqyb' %old idv: 1118115412, cc:2bl If $X$ is any quarter, and $Y$ is a $3$-cell that flanks it, then \\[ \\gamma(X,L) + \\gamma(Y,L) \\ge 0. \\] "; tags=[Marchal;Flypaper["OXLZLEZ";];Tex;Cfsqp;Xconvert;Branching;Split[0]]; };; add { idv = "FWGKMBZ"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus ); (&2 ,y2,sqrt8 ); (&2,y3,sqrt8); (&2,y4,sqrt8); (&2,y5,sqrt8 ); (&2,y6,sqrt8 ) ] (y_of_x delta_x y1 y2 y3 y4 y5 y6 > &0)`; doc = " This is used with rad2_x calculations to bound the denominator. "; tags=[Marchal;Flypaper["OXLZLEZ";"Nov2012"];Tex;Cfsqp;Xconvert;Branching]; };; add { idv = "BIXPCGW 9455898160"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2,y2, &2 * hminus); (&2,y3, &2 * hminus); (&2,y4, &2 * hminus); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] (gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun > -- #0.00569) `; doc = " OXLZLEZ.mod 'gamma_qu' If $X$ is a quarter, then $\\gamma(X,L)\\ge -0.00569$."; tags=[Marchal;Flypaper["OXLZLEZ";];Cfsqp;Clusterlp;Tex;Xconvert;Branching;Split[0]]; };; add { idv = "QITNPEA 5653753305"; ineq = all_forall `ineq [(&2 * hminus,y1, &2 * hplus); (&2,y2, &2 * hminus); (&2,y3, &2 * hminus); (&2,y4, &2 * hminus); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] (gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun + #0.0659 - #0.042*dih_y y1 y2 y3 y4 y5 y6 > #0.0)`; doc = " OXLZLEZ.mod 'azim1' This is an inequality for quarters used to prove the cluster inequality."; tags=[Marchal;Cfsqp;Flypaper["OXLZLEZ"];Clusterlp;Tex;Xconvert;Branching;Split[0]]; };; add { idv = "QITNPEA 6206775865"; ineq = all_forall `ineq [(&2 * hminus,y1, &2 * hplus); (&2,y2, &2 * hminus); (&2,y3, &2 * hminus); (#2 ,y4, &2 * hminus); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] (gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun + #0.0142852 - #0.00609451 *dih_y y1 y2 y3 y4 y5 y6 > #0.0) `; doc = " OXLZLEZ.mod 'gaz4' This is an inequality for quarters used to prove the cluster inequality."; tags=[Marchal;Cfsqp;Clusterlp;Flypaper["OXLZLEZ"];Tex;Xconvert;Branching;Split[0]]; };; add { idv = "QITNPEA 5814748276"; ineq = all_forall `ineq [(&2 * hminus,y1, &2 * hplus); (&2,y2, &2 * hminus); (&2,y3, &2 * hminus); (#2 ,y4, &2 * hminus); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] (gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun - #0.00127562 + #0.00522841 * dih_y y1 y2 y3 y4 y5 y6 > #0.0) `; doc = " OXLZLEZ.mod 'gaz5' This is an inequality for quarters used to prove the cluster inequality."; tags=[Marchal;Cfsqp;Clusterlp;Flypaper["OXLZLEZ"];Tex;Xconvert;Branching;Split[0]]; };; add { idv = "QITNPEA 3848804089"; ineq = all_forall `ineq [(&2 * hminus,y1, &2 * hplus); (&2,y2, &2 * hminus); (&2,y3, &2 * hminus); (#2 ,y4, &2 * hminus); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] (gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun - #0.161517 + #0.119482* dih_y y1 y2 y3 y4 y5 y6 > #0.0) `; doc = " OXLZLEZ.mod 'gaz6' This is an inequality for quarters used to prove the cluster inequality."; tags=[Marchal;Cfsqp;Clusterlp;Flypaper["OXLZLEZ"];Tex;Xconvert;Branching;Split[0]]; };; addtex(Section,"Packing -- Cluster Inequality","");; addtex(Comment,"", "\\claim{We show that the cluster inequality holds when there are at most two leaves along the critical edge.} Indeed, if the cluster has no quarter, then the inequality follows from \\ineq{GLFVCVK}. Assume that a quarter exists. Then there are two leaves, which flank the quarter. Next to the quarter is a three cell, because the dihedral angle of a $4$-cell is less than $\\pi$. The quarter and the $3$-cell both have weight $1$. The result follows from the given inequality. ");; (******************************************************************************) (* MARCHAL CELL_CLUSTER 3/4-COMBO CELL INEQS *) (******************************************************************************) add{ idv = "QITNPEA 5400790175 a" ; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2 ,y2, &2 *hminus); (&2 ,y3,&2 * hminus); (&2 * hminus,y4,sqrt8); (&2 ,y5,&2 * hminus); (&2 ,y6,&2 * hminus)] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun / &2 + beta_bump_lb > #0.0057) \/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2) \/ (eta_y y1 y2 y6 pow 2 < #1.34 pow 2))`; doc = " OXLZLEZ.mod 'gamma10' This is a $4$-cell (nonquarter) and adjacent $3$-cell inequality. The edges along the spine are small."; tags=[Marchal;Cfsqp;Flypaper["OXLZLEZ";];Clusterlp;Xconvert;Tex;Branching;Split[0;3]]; };; add{ idv = "QITNPEA 5400790175 b" ; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2 ,y2, &2 *hminus); (&2 ,y3,&2 * hminus); (&2 * hminus,y4,sqrt8); (&2 ,y5,&2 * hminus); (&2 ,y6,&2 * hminus)] ((gamma4fgcy y1 y2 y3 y4 y5 y6 lmfun / &2 + beta_bump_lb + gamma3f y1 y2 y6 sqrt2 lmfun > #0.0057) \/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 > &2) \/ (eta_y y1 y2 y6 pow 2 > #1.34 pow 2))`; doc = " OXLZLEZ.mod 'gamma11' This is a $4$-cell (nonquarter) and adjacent $3$-cell inequality. The edges along the spine are small."; tags=[Marchal;Cfsqp;Flypaper["OXLZLEZ";];Clusterlp;Xconvert;Tex;Branching;Split[0;3]]; };; (******************************************************************************) (* MARCHAL CELL_CLUSTER 2/3-CELLS QY FOR LEMMA OXLZLEZ *) (******************************************************************************) add { idv = "TEWNSCJ"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2,y2, &2 * hminus); (&2,y3, &2 * hminus); (&2,y4, sqrt8); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] ( (y_of_x (gamma23_full8_x (h0cut y1)) y1 y2 y3 y4 y5 y6 > a_spine5 + b_spine5 * dih_y y1 y2 y3 y4 y5 y6) \/ y_of_x rad2_x y1 y2 y3 y4 y5 y6 < &2 \/ eta_y y1 y2 y6 pow 2 > #1.34 pow 2 \/ eta_y y1 y3 y5 pow 2 > #1.34 pow 2 )`; doc = " 5 LEAF INEQ. Special case of ZTG...F23."; tags=[Marchal;Flypaper["OXLZLEZ";"Jun2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching;Split[0]]; };; add { idv = "PEMKWKU"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2,y2, sqrt8); (&2,y3, &2 * hminus); (&2,y4, sqrt8); (&2,y5, &2 * hminus); (&2,y6, sqrt8) ] ( (y_of_x (gamma23_keep135_x (h0cut y1)) y1 y2 y3 y4 y5 y6 > a_spine5 + b_spine5 * dih_y y1 y2 y3 y4 y5 y6) \/ y_of_x rad2_x y1 y2 y3 y4 y5 y6 < &2 \/ dih_y y1 y2 y3 y4 y5 y6 > #1.074 \/ eta_y y1 y2 y6 pow 2 > &2 \/ eta_y y1 y2 y6 pow 2 < #1.34 pow 2 \/ eta_y y1 y3 y5 pow 2 > #1.34 pow 2 )`; doc = "test"; tags=[Marchal;Flypaper["OXLZLEZ";"Jun2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching;Split[0]]; };; add { idv = "QITNPEAv2 4003532128"; ineq = all_forall `ineq [(&2 * hminus,y1, &2 * hplus); (&2,y2, &2 * hminus); (&2,y3, &2 * hminus); (&2 ,y4, sqrt8); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] (( eta_y y1 y2 y6 pow 2 > #1.34 pow 2 ) \/ (eta_y y1 y3 y5 pow 2 > #1.34 pow 2) \/ (y_of_x rad2_x y1 y2 y3 y4 y5 y6 < &2) \/ (y2 < y3) \/ (y2 < y5) \/ (y2 < y6) \/ (y_of_x (gamma23_full8_x (h0cut y1)) y1 y2 y3 y4 y5 y6 - #0.00457511 - #0.00609451*dih_y y1 y2 y3 y4 y5 y6 > #0.0) )`; doc = " OXLZLEZ.mod 'gaz3a' (Note the lower bound on $y_4$ is $2.1$. removed June 2012) This is an inequality for $23$-cells used to prove the cluster inequality. We may use monotonicity so that rad2 is exactly 2. By symmetry we may assume that y2 is the longest of y2,y3,y5,y6."; tags=[Marchal;Cfsqp;Cfsqp_branch 6; Clusterlp;Flypaper["OXLZLEZ";"Jun2012"];Tex;Xconvert;Branching;Split[0];] };; add{ idv = "QITNPEA 3725403817"; ineq = all_forall `ineq [(&2 * hminus,y1, &2 * hplus); (&2,y2, &2 * hminus); (&2,y3, &2 * hminus); (#2 ,y4, #2.1); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] (dih_y y1 y2 y3 y4 y5 y6 < #1.56) `; doc = " OXLZLEZ.mod 'azim3b' This is an inequality for $2$- and $3$-cells used to prove the cluster inequality. Note that $y_4\\le 2.1$. "; tags=[Marchal;Cfsqp;Clusterlp;Flypaper["OXLZLEZ"];Tex;Xconvert]; };; add { idv = "TXQTPVC"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2,y2, &2 * hminus); (&2,y3, &2 * hminus); (&2,y4, sqrt8); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] ( (y_of_x (gamma23_full8_x (h0cut y1)) y1 y2 y3 y4 y5 y6 > &3 * #0.0057) \/ y_of_x rad2_x y1 y2 y3 y4 y5 y6 < &2 \/ dih_y y1 y2 y3 y4 y5 y6 > #2.089 \/ dih_y y1 y2 y3 y4 y5 y6 < #1.946 \/ eta_y y1 y2 y6 pow 2 > #1.34 pow 2 \/ eta_y y1 y3 y5 pow 2 > #1.34 pow 2 )`; doc = "test. Is this needed? tch - 11/2012. Yes, June 22, 2012 notes page 1, in case 3 QX+1 QY"; tags=[Marchal;Flypaper["OXLZLEZ";"Jun2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching;Split[0]]; };; add { idv = "IXPOTPA"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2,y2, &2 * hminus); (&2,y3, &2 * hminus); (sqrt8,y4, &4 * hminus); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] (let tan2lower = #3.07 in ( // Tan[Pi-2.089]^2 let tan2upper = #6.45 in ( // Tan[Pi-1.946]^2 delta_y y1 y2 y3 y4 y5 y6 < &0 \/ delta4_y y1 y2 y3 y4 y5 y6 > &0 \/ (&4 * x1_delta_y y1 y2 y3 y4 y5 y6 < tan2lower * delta4_squared_y y1 y2 y3 y4 y5 y6) \/ (&4 * x1_delta_y y1 y2 y3 y4 y5 y6 > tan2upper * delta4_squared_y y1 y2 y3 y4 y5 y6) \/ eta_y y1 y2 y6 pow 2 > #1.34 pow 2 \/ eta_y y1 y3 y5 pow 2 > #1.34 pow 2 \/ (y_of_x (gamma23_full8_x (h0cut y1)) y1 y2 y3 y4 y5 y6 > &3 * #0.0057))))`; doc = "Dec 2, 2012. This is the case of TXQTPVC, when y4 >= sqrt8. We can't use monotonicty here, because of the explicit dih constraints."; tags=[Marchal;Flypaper["OXLZLEZ";"Nov2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching;Split[0];Cfsqp_branch 6;Penalty(50.0,500.0)]; };; add { idv = "QITNPEA 4003532128 a"; ineq = all_forall `ineq [(&2 * hminus,y1, &2 * hplus); (&2,y2, sqrt8); (sqrt2,y3,sqrt2); (sqrt2,y4,sqrt2); (sqrt2,y5,sqrt2); (&2,y6, sqrt8) ] ( (delta4_y y1 y2 y3 y4 y5 y6 > &25) \/ (delta_y y1 y2 y3 y4 y5 y6 > #0.14) \/ (delta_y y1 y2 y3 y4 y5 y6 < &0) )`; doc = " OXLZLEZ.mod 'gaz3a' This gives an upper bound $0.08$ on the dihedral angle of the $3$-cell, when delta < 0.14. This is an inequality for $23$-cells used to prove the cluster inequality."; tags=[Marchal;Cfsqp;Clusterlp;Flypaper["OXLZLEZ"];Tex;Xconvert;Branching]; (* lindih_lt: dih < 0.08 <=> 25*Tan[0.08] > Sqrt[4 (2hplus)^2 0.14] *) (* lindih_lt: dih < 0.037 <=> 25*Tan[0.037] > Sqrt[4 (2hplus)^2 0.03] *) (* 2012/6 corrected missing x1: was d4 > 25 > Tan[Pi/2 - 0.03] Sqrt[4 0.14] ==> dih <= 0.03. *) };; (******************************************************************************) (* MARCHAL 3-CELL NONNEGATIVITY *) (******************************************************************************) addtex(Section,"Marchal 3-cell nonnegativity","");; (* next series was added June 24, 2012 to replace old ZTG...F23 and related inequalities. Verifications are much faster now. *) add { idv = "RQWUDDU"; (* was "testdih" *) ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2,y2, sqrt8); (&2,y3, sqrt8); (&2,y4, sqrt8); (&2,y5, sqrt8); (&2,y6, sqrt8) ] ( dih_y y1 y2 y3 y4 y5 y6 > #0.76 \/ eta_y y1 y2 y6 pow 2 > &2 \/ eta_y y1 y3 y5 pow 2 > &2 )`; doc = "Min angle on a cell along a spine. Nov 2012: this gets used in jun 23, 2012 notes to reduce to at most 5 leaves."; tags=[Marchal;Flypaper["OXLZLEZ";"Jun2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching;Split[0]]; };; add { idv = "GCKBQEA"; ineq = all_forall `ineq [(&2 * hminus, y1, &2 * hplus); (&2,y2, sqrt8); (&2,y3, sqrt8); (&2,y4, sqrt8); (&2,y5, sqrt8); (&2,y6, sqrt8) ] ( dih_y y1 y2 y3 y4 y5 y6 > #0.606 )`; doc = "Min angle on a cell along a spine. added Dec 2, 2012"; tags=[Marchal;Flypaper["OXLZLEZ";"Nov2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching]; };; add { idv = "QZECFIC wt0"; ineq = all_forall `ineq [(&1,y1,&1); (&1,y2,&1); (&1,y3,&1); (#2.01,y4,&2 * hminus); (&2,y5, &2 *hminus); (&2,y6, &2 * hminus) ] ( y_of_x (gamma3f_x_div_sqrtdelta (&1) (&1) (&1)) y1 y2 y3 y4 y5 y6 > &0 )`; doc = "positivity of 3-cells on subcritical domain"; tags=[Marchal;Flypaper["OXLZLEZ";"Nov2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching]; };; add { idv = "QZECFIC wt0 corner"; ineq = all_forall `ineq [(&1,y1,&1); (&1,y2,&1); (&1,y3,&1); (&2,y4,#2.01); (&2,y5,#2.01); (&2,y6,#2.01) ] ( y_of_x (gamma3f_x_div_sqrtdelta (&1) (&1) (&1)) y1 y2 y3 y4 y5 y6 >= &0 )`; doc = "positivity of 3-cells on subcritical domain. Corner near sharp point"; tags=[Marchal;Flypaper["OXLZLEZ";"Nov2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching;Sharp;Eps 1.0e-8]; };; add { idv = "QZECFIC wt0 sqrt8"; ineq = all_forall `ineq [(&1,y1,&1); (&1,y2,&1); (&1,y3,&1); (&2 * hplus,y4,sqrt8); (&2,y5, &2 * hminus); (&2,y6, &2 * hminus) ] ( y_of_x (gamma3f_x_div_sqrtdelta (&0) (&1) (&1)) y1 y2 y3 y4 y5 y6 > &0 \/ eta_y y4 y5 y6 pow 2 > &2 )`; doc = "positivity of 3-cells on subcritical domain. The interval arithmetic calculation must kill off the '0'-branch of dih4, to avoid instability."; tags=[Marchal;Flypaper["OXLZLEZ";"Nov2012"];Widthcutoff 0.0002;Cfsqp;Clusterlp;Tex;Xconvert;Branching]; };; add { idv = "QZECFIC wt1"; (* was "test ratio" *) ineq = all_forall `ineq [(sqrt2,y1,sqrt2); (sqrt2,y2,sqrt2); (sqrt2,y3,sqrt2); (&2 * hminus, y4, &2 * hplus); (&2 ,y5, &2 * hminus); (&2 ,y6, &2 * hminus) ] ( y_of_x (gamma3f_x_div_sqrtdelta (h0cut y4) (&1) (&1)) y1 y2 y3 y4 y5 y6 > #0.008 * y_of_x dih4_x_div_sqrtdelta_posbranch y1 y2 y3 y4 y5 y6 \/ eta_y y4 y5 y6 pow 2 > &2 )`; doc = "gamma3f averages at least 0.008 per azim."; tags=[Marchal;Flypaper["OXLZLEZ";"Jun2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching;Split[3]]; };; add { idv = "QZECFIC wt2 A"; (* was "test ratio" , y4 y5 swapped, nov 2012, upper bounds on y4 y5 both made sqrt8 on Nov 27, 2012 *) ineq = all_forall `ineq [(sqrt2,y1,sqrt2); (sqrt2,y2,sqrt2); (sqrt2,y3,sqrt2); (&2 * hminus ,y4, sqrt8); (&2 * hminus, y5, sqrt8); (&2 ,y6, &2 * hminus) ] ( y_of_x (gamma3f_x_div_sqrtdelta (h0cut y4) (h0cut y5) (&1)) y1 y2 y3 y4 y5 y6 / &2 > #0.008 * y_of_x dih4_x_div_sqrtdelta_posbranch y1 y2 y3 y4 y5 y6 \/ eta_y y4 y5 y6 pow 2 > &2 )`; doc = "gamma3f averages at least 0.008 per azim. We don't have a wt3 case because eta[2hminus,2hminus,2hminus]>sqrt2."; tags=[Marchal;Flypaper["OXLZLEZ";"Jun2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching;Split[3;4]]; };; add { idv = "CIHTIUM"; ineq = all_forall `ineq [ (&1,y1,&1); (&1,y2,&1); (&1,y3,&1); (&2 * hminus,y4, sqrt8); (&2 * hminus ,y5, sqrt8); (&2 * hminus ,y6, sqrt8) ] (eta_y y4 y5 y6 pow 2 > &2 )`; doc = "every 3-cell has a subcritical edge"; tags=[Marchal;Flypaper["OXLZLEZ";"Nov2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching]; };; add { idv = "CJFZZDW"; (* y4 y6 swapped, nov 2012 *) ineq = all_forall `ineq [ (&1,y1,&1); (&1,y2,&1); (&1,y3,&1); (&2 * hplus ,y4, sqrt8); (&2 * hplus,y5, sqrt8); (&2,y6, sqrt8) ] (eta_y y4 y5 y6 pow 2 > &2 )`; doc = "no 3-cell has two supercritical edges"; tags=[Marchal;Flypaper["OXLZLEZ";"Nov2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching]; };; add { idv = "JSPEVYT"; ineq = all_forall `ineq [ (&1,y1,&1); (&1,y2,&1); (&1,y3,&1); (&2 * hminus,y4, sqrt8); (&2 * hminus ,y5, sqrt8); (&2 ,y6, sqrt8) ] (eta_y y4 y5 y6 pow 2 > (#1.34) pow 2 )`; doc = "eta small implies face small"; tags=[Marchal;Flypaper["OXLZLEZ";"Nov2012"];Cfsqp;Clusterlp;Tex;Xconvert;Branching]; };; (******************************************************************************) (* MARCHAL 2-CELL NONNEGATIVITY *) (******************************************************************************) addtex(Section,"Marchal 2-cell nonnegativity","");; add { idv = "GRKIBMP A V2"; ineq = all_forall `ineq [((&2),y1, (&2 * hplus)); (&1,y2,&1); (&1,y3,&1); (&1,y4,&1); (&1,y5,&1); (&1,y6,&1) ] (y_of_x (gamma2_x1_div_a_v2 (h0cut y1)) y1 y2 y3 y4 y5 y6 > #0.008 )`; doc = "gamma2 at least 0.008 per azim along critical edge."; tags=[Marchal;Flypaper["OXLZLEZ";"Jan2013"];Cfsqp;Clusterlp;Tex;Xconvert; Branching;Split[0]]; };; add { idv = "GRKIBMP B V2"; ineq = all_forall `ineq [((&2 * hplus),y1, (sqrt8)); (&1,y2,&1); (&1,y3,&1); (&1,y4,&1); (&1,y5,&1); (&1,y6,&1) ] (y_of_x (gamma2_x1_div_a_v2 (&0)) y1 y2 y3 y4 y5 y6 >= &0 )`; doc = "gamma2 nonnegative in general."; tags=[Marchal;Flypaper["OXLZLEZ";"Jan2013"];Cfsqp;Clusterlp;Tex;Xconvert;Sharp;Eps 1.0e-8; Branching;]; };; (******************************************************************************) (* MARCHAL CELL_CLUSTER 3Q1H INEQUALITIES FROM ineqdata3q1h.hl *) (******************************************************************************) (* Add all the inequalities from ineqdata3q1h.hl *) let add3q1h = let records = Ineqdata3q1h.records in let mk_ineq = Ineqdata3q1h.mk_ineq in let mk_3q1h case r = let d = List.nth records r in let f = mk_ineq case d in { ineq = f; idv = Printf.sprintf "OXLZLEZ 6346351218 %d %d" case r; tags=(if (case,r)=(0,0) then [Tex] else []) @ [Marchal;Clusterlp;Branching;Flypaper["OXLZLEZ";]] @ (if (case >0) then [Xconvert] else []); doc = "One of many inequalities for four leaves, three quarters, and simplex of weight $1/2$"; } in let mk_3q1hc r = map (fun c -> mk_3q1h c r) (0-- 4) in let mk_3q1h_all = List.flatten(map (mk_3q1hc) (0--((List.length records)-1))) in map add mk_3q1h_all;; (******************************************************************************) (* PACKING COUNTING SPHERES INEQUALITIES *) (******************************************************************************) addtex (Section,"Packing -- Polyhedra","");; add{ idv = "1965189142 34"; tags = [Tex;Flypaper["BIEFJHU"]]; doc = "This inequality gives a linear lower bound on the area of a regular spherical polygon, when $k\\ge 34$"; ineq = all_forall `ineq [ (#1.0,x1,#1.26); (&1,x2,&1);(&1,x3,&1);(&1,x4,&1);(&1,x5,&1);(&1,x6,&1) ] (#0.591 - #0.0331 * &34 + #0.506 * lfun_y1 x1 x2 x3 x4 x5 x6 < &0)` };; add{ idv = "1965189142 a"; tags = [Tex;Flypaper["BIEFJHU"]]; doc = " %old idv: 7991525482, eqn:alin, This inequality gives a linear lower bound on the area of a regular spherical polygon, when $k\\le 34$. Let $L$ be given by Definition~\\ref{def:L}. Let \\[ g(h) = \\arccos(h/2) - \\pi/6. \\] Let \\[ \\op{reg}(a,k) = 2\\pi - 2 k (\\arcsin(\\cos(a)\\sin(\\pi/k))). \\] Then \\[ \\op{reg}(g(h),k) \\ge c_0 + c_1 k + c_2 L(h),\\quad k = 3,4,\\ldots,\\quad 1\\le h\\le \\hm. \\] The integer parameter $k$ has been replaced with a real variable. If $k\\ge34$, then the right-hand-side is negative and the inequality is immediate. The trig identity \\[ \\cos(\\acos (h/2) - \\pi/6)) = h \\sqrt(3)/4 + \\sqrt(1-(h/2)^2)/2. \\] has been used to simplify the inequality. "; ineq = all_forall `ineq [ (#1.0,x1,#1.26); (#3.0,x2,#34.0);(&1,x3,&1);(&1,x4,&1);(&1,x5,&1);(&1,x6,&1) ] (&2 * pi - &2 * asnFnhk x1 x2 x3 x4 x5 x6 > #0.591 - #0.0331 * x2 + #0.506 * lfun_y1 x1 x2 x3 x4 x5 x6)` };; add{ idv = "8055810915"; tags = [Tex;Flypaper["WAZLDCD"]]; doc = "This inequality gives the nonoverlap of disks on the unit sphere."; ineq = all_forall `ineq [ (&4, x1, #2.52 pow 2); (&4,x2,&4); ((&2 * h0) pow 2,x3,(&2 * h0) pow 2); (&1,x4,&1); (&1,x5,&1); (&1,x6,&1) ] (acs_sqrt_x1_d4 x1 x2 x3 x4 x5 x6 - pi/ (&6) + #0.797 < arclength_x_123 x1 x2 x3 x4 x5 x6)`; };; add{ idv = "6096597438 a"; tags = [Tex;Flypaper["UKBRPFE"]]; doc = "This inequality gives a linear lower bound on the area of a regular spherical polygon, when $k\\ge 64$"; ineq = all_forall `ineq [ (#1.0,h,#1.0) ] (#0.591 - #0.0331 * &64 + #0.506 * lfun (&1) + #1.0 < &0)` };; add{ idv = "6096597438 b"; tags = [Tex;Flypaper["UKBRPFE"]]; doc = " % old idv cc:alin2. 8540377696 Let \\[ g(h) = \\arccos(h/2) - \\pi/6. \\] Let \\[ \\op{reg}(a,k) = 2\\pi - 2 k (\\arcsin(\\cos(a)\\sin(\\pi/k))). \\] Let \\[ a'=0.797\\approx \\arc(2,2,2\\hm)-g(\\hm).\\] Then for $k=3,~4$,\\dots \\[ \\op{reg}(a',k) \\ge c_0 + c_1 k + c_2 L(1) +c_3.\\] This inequality gives a linear lower bound on the area of a regular spherical polygon, when $k\\le 64$. The integer parameter $k$ has been replaced with a real variable. If $k\\ge64$, then the right-hand-side is negative and the inequality is immediate. "; ineq = all_forall `ineq [ (#3.0,x1,#64.0); (&1,x2,&1); (&1,x3,&1); (&1,x4,&1); (&1,x5,&1); (&1,x6,&1) ] (&2 * pi - &2 * asn797k x1 x2 x3 x4 x5 x6 > #0.591 - #0.0331 * x1 + #0.506 * lfun (&1) + #1.0)` };; (******************************************************************************) (* LOCAL FAN INEQUALITIES *) (******************************************************************************) addtex(Section,"Local Fan -- Standard Fan","");; add{ idv = "4717061266"; ineq = all_forall `ineq [ (&2, y1, &2*h0); (&2, y2, &2*h0); (&2, y3, &2*h0); (&2, y4, &2*h0); (&2, y5, &2*h0); (&2, y6, &2*h0) ] (delta_y y1 y2 y3 y4 y5 y6 > &0)`; tags = [Flypaper["TVAWGDR"];Main_estimate;Tex;Xconvert]; doc = "A certain configuration of four points cannot be coplanar. Note 2013-06-27: This doesn't ever get used. I think it can be skipped. 2013-08-05. Now used in the main estimate. It could be moved there. "; };; addtex(Section,"Local Fan"," -- Minimal Fan");; add{ idv = "SDCCMGA a"; doc = "This is an arclength estimate. 2013-06-24. This does not seem to be used anywhere."; tags = [Tex;Flypaper["SDCCMGA"];Cfsqp;Xconvert]; ineq = all_forall `ineq [ (&2,y1,#2.52); (&1,y2,&1);(&1,y3,&1);(&1,y4,&1);(&1,y5,&1);(&1,y6,&1) ] (arclength_y1 (&2) (&2 * h0) y1 y2 y3 y4 y5 y6 + arclength_y1 (&2) (&2 * h0) y1 y2 y3 y4 y5 y6 < arclength_y1 (&2*h0) (&2 * h0) y1 y2 y3 y4 y5 y6 + &2 * arc_hhn)`; };; add{ idv = "SDCCMGA b"; doc = "This is an arclength estimate. 2013-06-24. This does not seem to be used anywhere."; tags = [Tex;Flypaper["SDCCMGA"];Cfsqp;Xconvert]; ineq = all_forall `ineq [ (&2,y1,#2.52); (&1,y2,&1);(&1,y3,&1);(&1,y4,&1);(&1,y5,&1);(&1,y6,&1) ] (arclength_y1 (&2) (&2 * h0) y1 y2 y3 y4 y5 y6 + arclength_y1 (&2) (&2 * h0) y1 y2 y3 y4 y5 y6 < arclength_y1 (&2*h0) (&2) y1 y2 y3 y4 y5 y6 + pi / &3 + arc_hhn)`; };; add{ idv = "JNTEFVP 1"; doc = "A quad with a reflex vertex has a diagonal less than $\\sqrt8$. This allows us to split a quad into two simplices. By extreme edge we can assume the diagonal is $2h_0$ or $\\sqrt8$. The case $2h_0$ is already done with the triangles."; tags=[Tex;Cfsqp;Flypaper["JNTEFVP"]]; ineq = all_forall `ineq [ (&4,x1,(&2 * h0) pow 2); (&4,x2,(&2 * h0) pow 2); (&4,x3,(&2 * h0) pow 2); (&4,x4,(&2 * h0) pow 2); (&4,x5,(&2 * h0) pow 2); (&8 ,x6, (&4 * h0) pow 2) ] (delta_x4 x1 x2 x3 x4 x5 x6 > &0)`; };; addtex (Section,"Tame Hypermap","");; add{ idv = "4652969746 1"; doc = "This is the calculation of the (p,q)=(5,0) entry in tame table b."; tags=[Cfsqp;Tex;Tablelp;Flypaper["KCBLRQC"];Xconvert]; ineq = all_forall `ineq [ (&2,y1,#2.52); (&2,y2,#2.52); (&2,y3,#2.52); (#2.1771,y4,#2.52); (&2,y5,#2.52); (&2,y6,#2.52) ] (taum y1 y2 y3 y4 y5 y6 > #0.04)`; };; add{ idv = "4652969746 2"; doc = "This is the calculation of the (p,q)=(5,0) entry in tame table b."; tags=[Cfsqp;Tex;Tablelp;Flypaper["KCBLRQC"];Xconvert]; ineq = all_forall `ineq [ (&2,y1,#2.52); (&2,y2,#2.52); (&2,y3,#2.52); (&2,y4,#2.1771); (&2,y5,#2.52); (&2,y6,#2.52) ] (taum y1 y2 y3 y4 y5 y6 - #0.312 * (dih_y y1 y2 y3 y4 y5 y6 - &2 * pi/ &5) > #0.04 / &5)`; };; (* interval arithmetic bounds DART4 *) add{ idv = "2570626711"; doc=""; tags = [Cfsqp;Tablelp;Flypaper["KCBLRQC"];Xconvert;Tex]; ineq = all_forall `ineq [ (#2.0,y1,&2 * h0); (#2.0,y2,&2 * h0); (#2.0,y3,&2 * h0); (&2 * h0,y4,&2 * h0); (#2.0,y5,&2 * h0); (#2.0,y6,&2 * h0)] (dih_y y1 y2 y3 y4 y5 y6 > #1.15)`; };; (******************************************************************************) (* LINEAR PROGRAM INEQUALITIES *) (******************************************************************************) addtex(Section,"Linear Programs","");; let dart_classes = ref [];; let define_dart t = let th = new_definition t in let _ = (dart_classes := th :: (!dart_classes)) in th;; (* The bounds on the four vertices $\\v_1,\\ldots,\\v_4$ gives the top simplex bound \\[ y_4 \\le \\op{edge\\_flat}(2h_0,2h_0,2h_0,2h_0,2h_0) < 4.37 \\ *) add{ idv = "3287695934"; doc=""; tags = [Cfsqp;Tablelp;Xconvert;Tex]; ineq = all_forall `ineq [ (#4.37,y1,&4 * h0); (#2.0,y2,&2 * h0); (#2.0,y3,&2 * h0); (&2 * h0,y4,&4 * h0); (#2.0,y5,&2 * h0); (#2.0,y6,&2 * h0)] (delta_y y1 y2 y3 y4 y5 y6 < &0)`; };; let dart_std4 = define_dart `dart_std4 y1 y2 y3 y4 y5 y6 y7 y8 y9 = [ (#2.0,y1,&2 * h0); (#2.0,y2,&2 * h0); (#2.0,y3,&2 * h0); (&2 * h0,y4,#4.37); (#2.0,y5,&2 * h0); (#2.0,y6,&2 * h0); (#2.0,y7,&2 * h0); (#2.0,y8,&2 * h0); (#2.0,y9,&2 * h0)]`;; skip{ idv = "8673686234"; doc="Special Nonlinear Inequality for Lp. This is not autogenerated. $y_1$ is the shorter diagonal. Ran for 95 million steps and it didn't pass. Replaced with -a, -b, -c below."; tags = [Cfsqp;Tex;Xconvert;Deprecated]; ineq = all_forall `ineq [ (sqrt8,y1,#3.0); (&2,y2,&2 * h0); (&2,y3,&2 * h0); (sqrt8,y4,&4 * h0); (&2,y5,&2 * h0); (&2,y6,&2 * h0) ] ((y2 + y3 + y5 + y6 - #7.99 > #2.75 * (y1 - sqrt8)) \/ (delta_y y1 y2 y3 y4 y5 y6 < &0) \/ (y4 < y1)) `; };; add{ idv = "8673686234 a"; doc="Special Nonlinear Inequality for Lp. This is not autogenerated. See head.mod. $y_1$ is the shorter diagonal. Cut out a piece near {sqrt8,2,2,sqrt8,2,2}."; tags = [Cfsqp;Lp_aux "8673686234";Tex;Xconvert]; ineq = all_forall `ineq [ (sqrt8,y1,#3.0); (&2,y2,#2.07); (&2,y3,#2.07); (sqrt8,y4,&4 * h0); (&2,y5,#2.07); (&2,y6,#2.07) ] ((y2 + y3 + y5 + y6 - #7.99 - #0.00385 * delta_y y1 y2 y3 y4 y5 y6 > #2.75 * ((y1 + y4)/ &2 - sqrt8)) ) `; };; add{ idv = "8673686234 b"; doc="Special Nonlinear Inequality for Lp. This is not autogenerated. $y_1$ is the shorter diagonal. This is the case $y_4\\ge 3$. Stretch $y_4$ until $\\Delta=0$, then contract $y_4$ within the plane until $y_4=3$."; tags = [Cfsqp;Lp_aux "8673686234";Tex;Xconvert]; ineq = all_forall `ineq [ (sqrt8,y1,#3.0); (#2.07,y2,&2 * h0); (&2,y3,&2 * h0); (#3.0,y4,#3.0); (&2,y5,&2 * h0); (&2,y6,&2 * h0) ] ((y2 + y3 + y5 + y6 - #7.99 > #2.75 * (y1 - sqrt8)) \/ (delta_y y1 y2 y3 y4 y5 y6 < &0) ) `; };; add{ idv = "8673686234 c"; doc="Special Nonlinear Inequality for Lp. This is not autogenerated. $y_1$ is the shorter diagonal. This case takes $y_4\\le 3$."; tags = [Cfsqp;Lp_aux "8673686234";Tex;Xconvert]; ineq = all_forall `ineq [ (sqrt8,y1,#3.0); (#2.07,y2,&2 * h0); (&2,y3,&2 * h0); (sqrt8,y4,#3.0); (&2,y5,&2 * h0); (&2,y6,&2 * h0) ] ((y2 + y3 + y5 + y6 - #7.99 > #2.75 * ((y1 + y4)/ &2 - sqrt8)) \/ (y2 + y3 + y5 + y6 - #7.99 - #0.00385 * delta_y y1 y2 y3 y4 y5 y6 > #2.75 * ((y1 + y4)/ &2 - sqrt8)) \/ (delta_y y1 y2 y3 y4 y5 y6 < &0) ) `; };; add { idv="6170936724"; doc="Special nonlinear inequality for LP. Used to help prove 8673686234. Added 2013-06-17."; tags=[Cfsqp;Xconvert;Tex;Penalty(50.0,500.0)]; ineq = all_forall `ineq [ (&3,y1,&3); (&2,y2,#2.52); (&2,y3,#2.52); (sqrt8,y4,&4 * h0); (&2,y5,#2.52); (&2,y6,#2.52) ] ( y_of_x delta_x1 y1 y2 y3 y4 y5 y6 < &0)`; };; addtex(Comment,"Quad convexity justification"," We can use dimension reduction methods to reduce the number of variables. This is the reduced version that occurs when the cross diagonal is minimal. This reduction allows nonconvex deformations. The Dim_red_backsym extremalizes the edges x8 and x9. However, if the cross-diag $\\le \\sqrt8$, then by geometric considerations, the cross-diagonal is interior to the quad, and the subdivision is justified. ");; add{ idv = "7043724150 a"; doc="We can use dimension reduction methods to reduce the number of variables. Deprecated 2013-07-29"; (* \/ (delta_y y4 y9 y6 sqrt8 y5 y8 < &0) *) tags = [Cfsqp;Tablelp;Flypaper["KCBLRQC"];Lp;Xconvert;Tex;Dim_red_backsym;Quad_cluster 0.001]; ineq =all_forall `ineq (dart_std4 y1 y2 y3 y4 y5 y6 y7 y8 y9) (( tauq y1 y2 y3 y4 y5 y6 y7 y8 y9 + #4.72 * dih_y y1 y2 y3 y4 y5 y6 - #6.248 > #0.0) \/ ( enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9 < sqrt8 ) )`; };; add{ idv = "7043724150 a reduced v2"; doc="We can use dimension reduction methods to reduce the number of variables. This is the reduced version that occurs when the cross diagonal is minimal. See 'Quad convexity justification' comment. Corrected 2013-05-01, A.Solovyev / thales, y5 upper bound was &2 * h0. Deprecated 2013-07-29. "; tags = [Cfsqp;Lp_aux "7043724150 a";Xconvert;Tex;]; ineq =all_forall `ineq [ (&2,y1,&2 * h0); (&2,y2,&2 * h0); (&2,y3,&2 * h0); (&2,y4,&2 * h0); (&2 * h0,y5,sqrt8); (&2,y6,&2 * h0) ] ( taum y1 y2 y3 y4 y5 y6 + #4.72 * dih_y y1 y2 y3 y4 y5 y6 - #6.248 / &2 > #0.0)`; };; add { idv = "6944699408 a"; doc="We can use dimension reduction methods to reduce the number of variables. Deprecated 2013-07-29."; tags = [Cfsqp;Tablelp;Flypaper["KCBLRQC"];Lp;Xconvert;Tex;Dim_red_backsym;Quad_cluster 0.0005]; ineq = all_forall `ineq (dart_std4 y1 y2 y3 y4 y5 y6 y7 y8 y9) (( tauq y1 y2 y3 y4 y5 y6 y7 y8 y9 + #0.972 * dih_y y1 y2 y3 y4 y5 y6 - #1.707 > #0.0) \/ ( enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9 < sqrt8 ))`; };; add { idv = "6944699408 a reduced"; doc="We can use dimension reduction methods to reduce the number of variables. This is the reduced version that occurs when the cross diagonal is minimal. See 'Quad convexity justification' comment. Deprecated 2013-07-29. " ; tags = [Cfsqp;Lp_aux "6944699408 a";Xconvert;Tex]; ineq = all_forall `ineq [ (&2,y1,&2 * h0); (&2,y2,&2 * h0); (&2,y3,&2 * h0); (&2,y4,&2 * h0); (&2 * h0,y5,sqrt8); (&2,y6,&2 * h0) ] ( taum y1 y2 y3 y4 y5 y6 + #0.972 * dih_y y1 y2 y3 y4 y5 y6 - #1.707 / &2 > #0.0)`; };; (* constant 1.433 corrected 2010-06-14 *) add{ idv = "4240815464 a"; doc=""; tags = [Cfsqp;Tablelp;Flypaper["KCBLRQC"];Tex;Xconvert;Lp;Penalty(10000.0,500.0);Dim_red_backsym;Quad_cluster 0.0001]; ineq = all_forall `ineq (dart_std4 y1 y2 y3 y4 y5 y6 y7 y8 y9) (( tauq y1 y2 y3 y4 y5 y6 y7 y8 y9 + #0.7573 *dih_y y1 y2 y3 y4 y5 y6 - #1.433 > #0.0) \/ ( enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9 < sqrt8 ) \/ ( delta_y y1 y2 y3 y4 y5 y6 < &0) \/ ( delta_y y7 y2 y3 y4 y8 y9 < &0)) `; };; add{ idv = "4240815464 a reduced"; doc="We can use dimension reduction methods to reduce the number of variables. This is the reduced version that occurs when the cross diagonal is minimal. See 'Quad convexity justification' comment. "; tags = [Cfsqp;Lp_aux "4240815464 a";Tex;Xconvert]; ineq = all_forall `ineq [ (&2,y1,&2 * h0); (&2,y2,&2 * h0); (&2,y3,&2 * h0); (&2,y4,&2 * h0); (&2 * h0,y5,sqrt8); (&2,y6,&2 * h0) ] ( taum y1 y2 y3 y4 y5 y6 + #0.7573 *dih_y y1 y2 y3 y4 y5 y6 - #1.433 / &2 > #0.0)`; };; add{ idv = "3862621143 revised"; doc="Revised May 24, 2011 for speed by extending enclosed to 3.01 and y4 to 2.9. See 'Quad convexity justification' comment."; tags = [Cfsqp;Tablelp;Flypaper["KCBLRQC"];Tex;Xconvert;Lp;Penalty(10000.0,500.0);Dim_red_backsym;Quad_cluster 0.05]; (* was Quad_cluster 0.0005 *) ineq = all_forall `ineq (dart_std4 y1 y2 y3 y4 y5 y6 y7 y8 y9) (( tauq y1 y2 y3 y4 y5 y6 y7 y8 y9 - #0.453 * dih_y y1 y2 y3 y4 y5 y6 + #0.777 > #0.0) \/ ( enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9 < #3.01 ) \/ ( delta_y y1 y2 y3 y4 y5 y6 < &0) \/ (y4 < #2.9) \/ ( delta_y y7 y2 y3 y4 y8 y9 < &0)) `; };; add { idv = "3862621143 side"; doc= "Do side splits out to 3.01."; tags = [Cfsqp;Lp_aux "3862621143 revised";Tex;Xconvert]; ineq = all_forall `ineq [ (&2,y1,&2 * h0); (&2,y2,&2 * h0); (&2,y3,&2 * h0); (&2,y4,&2 * h0); (&2 * h0,y5,#3.01); (&2,y6,&2 * h0) ] ( taum y1 y2 y3 y4 y5 y6 - #0.453 * dih_y y1 y2 y3 y4 y5 y6 + #0.777 / &2 > #0.0)`; };; add { idv = "3862621143 front"; doc= "Do front half when y4 <= 2.9 "; tags = [Cfsqp;Lp_aux "3862621143 revised";Tex;Xconvert]; ineq = all_forall `ineq [ (&2,y1,&2 * h0); (&2,y2,&2 * h0); (&2,y3,&2 * h0); (&2 * h0,y4,#2.9); (&2 ,y5,&2 * h0); (&2,y6,&2 * h0) ] ( taum y1 y2 y3 y4 y5 y6 + tame_table_d 2 1 - #0.453 * dih_y y1 y2 y3 y4 y5 y6 + #0.777 > #0.0)`; };; add { idv = "3862621143 back"; doc= " Back half when y4 <= 2.0. We know taum > tame_table_d 2 1, when y4 in [sqrt8,3.01]. This extends a bit further. We could optimize by extremizing y4,y5,y6 and using known triangle calculations."; tags = [Cfsqp;Lp_aux "3862621143 revised";Tex;Xconvert]; ineq = all_forall `ineq [ (&2,y1,&2 * h0); (&2,y2,&2 * h0); (&2,y3,&2 * h0); (sqrt8,y4,#3.01); (&2 ,y5,&2 * h0); (&2,y6,&2 * h0) ] ( taum y1 y2 y3 y4 y5 y6 > tame_table_d 2 1)`; };; skip { idv = "3862621143 a"; doc="This was replaced with '3862621143 revised'. The revision is mathematically equivalent, but runs about 6 hours faster."; tags = [Cfsqp;Tablelp;Flypaper["KCBLRQC"];Tex;Xconvert;Lp;Penalty(10000.0,500.0);Dim_red_backsym;Quad_cluster 0.0005;Deprecated]; ineq = all_forall `ineq (dart_std4 y1 y2 y3 y4 y5 y6 y7 y8 y9) (( tauq y1 y2 y3 y4 y5 y6 y7 y8 y9 - #0.453 * dih_y y1 y2 y3 y4 y5 y6 + #0.777 > #0.0) \/ ( enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9 < sqrt8 ) \/ ( delta_y y1 y2 y3 y4 y5 y6 < &0) \/ ( delta_y y7 y2 y3 y4 y8 y9 < &0)) `; };; skip { idv = "3862621143 a reduced"; doc= "This has also been revised. See 'Quad convexity justification' comment."; tags = [Cfsqp;Lp_aux "3862621143 a";Tex;Xconvert;Deprecated]; ineq = all_forall `ineq [ (&2,y1,&2 * h0); (&2,y2,&2 * h0); (&2,y3,&2 * h0); (&2,y4,&2 * h0); (&2 * h0,y5,sqrt8); (&2,y6,&2 * h0) ] ( taum y1 y2 y3 y4 y5 y6 - #0.453 * dih_y y1 y2 y3 y4 y5 y6 + #0.777 / &2 > #0.0)`; };; add{ idv = "5691615370"; doc="This is one of the hand-entered LP inequalities (perimZ) in head.mod. \\claim{If a quadrilateral standard region has both diagonals at least $3$, then the perimeter is at least $8.472$.} Otherwise, some configuration has perimeter $\\le8.472$. We may disregard the origin $\\orz$ and show the result holds for $\\v_1,\\ldots,\\v_4$. We may increase a diagonal, with fixed perimeter, until the four points are coplanar. We may contract the diagonals until both equal $3$. This calculation provides the desired inequality. May 25, 2011. Symmetry added. Wlog, y2 is the longest edge among y2,y3,y5,y6. Wlog, y3 is the longest among y3,y6. "; tags = [Cfsqp;Lp_aux "5691615370";Lp_aux "9563139965D";Main_estimate;Tex;Xconvert]; ineq = all_forall `ineq [ (#3.0,y1,#3.0); (&2,y2,#2.52); (&2,y3,#2.52); (#3.0,y4,#3.0); (&2,y5,#2.52); (&2,y6,#2.52) ] ((delta_y y1 y2 y3 y4 y5 y6 < &0) \/ (y2 + y3 + y5 + y6 > #8.472) \/ (y2 < y3) \/ (y2 < y5) \/ (y2 < y6 ) \/ (y3 < y6))`; };; skip { idv="3748250573"; doc="Used with 5691615370. Added 2013-06-19."; tags=[Cfsqp;Xconvert;Lp_aux "5691615370";Tex;Penalty(50.0,500.0)]; ineq = all_forall `ineq [ (&2 * h0,y1,&4 * h0); (&2,y2,#2.52); (&2,y3,#2.52); (&2 * h0,y4,&4 * h0); (&2,y5,#2.52); (&2,y6,#2.52) ] ( y_of_x ups_126 y1 y2 y3 y4 y5 y6 > &0 \/ delta_y y1 y2 y3 y4 y5 y6 < &0 )`; };; add { idv="5584033259"; doc= "Used with 5691615370. Added 2013-06-19."; tags=[Cfsqp;Xconvert;Tex;Lp_aux "5691615370";Penalty(50.0,500.0)]; ineq = all_forall `ineq [ (&3,y1,&4 * h0); (&2,y2,#2.472); (&2,y3,#2.472); (&3,y4,&4 * h0); (&2,y5,#2.472); (&2,y6,#2.472) ] ( y1 < &4 \/ delta_y y1 y2 y3 y4 y5 y6 < &0 )`; };; (* Second derivative (2986512...) dimension reduction is used on quad to reduce $y_4$ (the diagonal) to 3. *) (* let dart4_diag3_a = define_dart `dart4_diag3_a y1 y2 y3 y4 y5 y6 y7 y8 y9 = [ (&2,y1,#2.52); (&2,y2,#2.52); (&2,y3,#2.52); (&3,y4,&2 * #2.52); (&2,y5,#2.52); (&2,y6,#2.52); (&2,y7,#2.52); (&2,y8,#2.52); (&2,y9,#2.52) ]`;; *) let dart4_diag3_b = define_dart `dart4_diag3_b y1 y2 y3 y4 y5 y6 y7 y8 y9 = [ (&2,y1,#2.52); (&2,y2,#2.52); (&2,y3,#2.52); (&3,y4,&3); (&2,y5,#2.52); (&2,y6,#2.52); (&2,y7,#2.52); (&2,y8,#2.52); (&2,y9,#2.52) ]`;; (* tchales 2013-05-03, upper bound on y4 should be &2* #2.52. It is inconsequential because ineq "9563139965D" is not used formally. *) (* We use "add" rather than "skip" on 9563139965D to make it accessible to the linear programs. However, it is derived from other inequalities. Let's start with the ad hoc inequality: 9563139965D. (>= 0.467 ) By top edge contraction arguments, we may assume that (0) all top edges have length 2, or (1) both diagonals have length 3 (by contracting in different ways). The first case (0) is impossible by geomeric considerations: edges=2 ==> some diagonal <= sqrt8. So both diagonals have length 3. This has been completely solved in a series 9563139965D in ineq.hl. *) add{ idv = "9563139965D"; doc = "This is the special Lp estimate for quadrilaterals with both diagonals greater than 3.0. Dimension reduction can be used to simplify. Derived from 5691615370 and d,e,f below"; tags = [Cfsqp;Xconvert;Lp;Tex;Penalty(5.0,500.0);Derived ["d e f"]]; ineq = all_forall `ineq (dart4_diag3_b y1 y2 y3 y4 y5 y6 y7 y8 y9) ((tauq y1 y2 y3 y4 y5 y6 y7 y8 y9 > #0.467) \/ (enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9 < &3 ))`; };; (* Change to 9563139965, change the constant 0.496 to 0.467, then use dimension reduction below to prove with 5D calcs. Dec 18, 2010. *) add { idv = "9563139965 d"; doc = "This is the special Lp estimate for quadrilaterals with both diagonals greater than 3.0. Dimension reduction can be used to simplify. This is the reduced case with both diagonals 3.0. The correction term is based in ineq (5691615...). It holds for both simplices if the appropriate diagonal is selected. This is the case where all top edges of quad are at most 2.472."; tags = [Cfsqp;Lp_aux "9563139965D";Main_estimate;Xconvert;Tex]; ineq = all_forall `ineq [ (&2,y1,&2 * h0); (&2,y2,&2 * h0); (&2,y3,&2 * h0); (&3,y4,&3); (&2,y5,#2.472); (&2,y6,#2.472) ] ((taum y1 y2 y3 y4 y5 y6 + #0.5 * ( #8.472 / &2 - y5 - y6) > #0.467 / &2))`; };; add { idv = "9563139965 e"; doc = "This is the secondary estimate for quadrilaterals with both diagonals greater than 3.0. Dimension reduction can be used to simplify. This is the reduced case with both diagonals 3.0. This is the case where some top edges is at least 2.472. First triangle of split."; tags = [Cfsqp;Lp_aux "9563139965D";Main_estimate;Xconvert;Tex]; ineq = all_forall `ineq [ (&2,y1,&2 * h0); (&2,y2,&2 * h0); (&2,y3,&2 * h0); (&3,y4,&3); (#2.467,y5,&2 * h0); (&2,y6,&2 * h0) ] ((taum y1 y2 y3 y4 y5 y6 > #0.467 - #0.115))`; };; add { idv = "9563139965 f"; doc = "This is the secondary estimate for quadrilaterals with both diagonals greater than 3.0. Dimension reduction can be used to simplify. This is the reduced case with both diagonals 3.0. This is the case where some top edges is at least 2.472. Second triangle of split."; tags = [Cfsqp;Lp_aux "9563139965D";Main_estimate;Xconvert;Tex]; ineq = all_forall `ineq [ (&2,y1,&2 * h0); (&2,y2,&2 * h0); (&2,y3,&2 * h0); (&3,y4,&3); (&2,y5,&2 * h0); (&2,y6,&2 * h0) ] ((taum y1 y2 y3 y4 y5 y6 > #0.115))`; };; addtex(Section,"Linear Programs"," -- dart\_ std3");; let dart_std3 = define_dart `dart_std3 y1 y2 y3 y4 y5 y6 = [ (#2.0,y1,#2.52); (#2.0,y2,#2.52); (#2.0,y3,#2.52); (#2.0,y4,#2.52); (#2.0,y5,#2.52); (#2.0,y6,#2.52)]`;; (* Tame table B inequalities *) add{ idv = "5735387903"; doc=""; tags = [Cfsqp;Tablelp;Flypaper["KCBLRQC"];Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3 y1 y2 y3 y4 y5 y6) (dih_y y1 y2 y3 y4 y5 y6 > #0.852)`; };; (* changed from #1.893, to #1.9, for reasons I forget, thales, 2010-02-28, back again 2010-06-15. *) add{ idv = "5490182221"; doc=""; tags = [Cfsqp;Tablelp;Flypaper["KCBLRQC"];Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3 y1 y2 y3 y4 y5 y6) (dih_y y1 y2 y3 y4 y5 y6 < #1.893)`; };; (* let tame_hypermap_list = map (fun t -> (getexact t).ineq) ["5735387903";"5490182221";"2570626711"];; *) add{ idv = "3296257235"; doc=""; tags = [Cfsqp;Tablelp;Flypaper["KCBLRQC"];Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3 y1 y2 y3 y4 y5 y6) (taum y1 y2 y3 y4 y5 y6 + #0.626 * dih_y y1 y2 y3 y4 y5 y6 - #0.77 > #0.0)`; };; add{ idv = "8519146937"; doc=""; tags = [Cfsqp;Tablelp;Flypaper["KCBLRQC"];Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3 y1 y2 y3 y4 y5 y6) ( taum y1 y2 y3 y4 y5 y6 - #0.259 * dih_y y1 y2 y3 y4 y5 y6 + #0.32 > #0.0)`; };; add{ idv = "4667071578"; doc=""; tags = [Cfsqp;Tablelp;Flypaper["KCBLRQC"];Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3 y1 y2 y3 y4 y5 y6) ( taum y1 y2 y3 y4 y5 y6 - #0.507 * dih_y y1 y2 y3 y4 y5 y6 + #0.724 > #0.0)`; };; add{ idv = "1395142356"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3 y1 y2 y3 y4 y5 y6) ( taum y1 y2 y3 y4 y5 y6 + #0.001 - #0.18 * (y1 + y2 + y3 - #6.0) - #0.125 * (y4 + y5 + y6 - #6.0) > #0.0)`; };; add{ idv = "7394240696"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3 y1 y2 y3 y4 y5 y6) ( sol_y y1 y2 y3 y4 y5 y6 - #0.55125 - #0.196 * (y4 + y5 + y6 - #6.0) + #0.38 * (y1 + y2 + y3 - #6.0) > #0.0)`; };; add{ idv = "7726998381"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3 y1 y2 y3 y4 y5 y6) ( -- sol_y y1 y2 y3 y4 y5 y6 + #0.5513 + #0.3232 * (y4 + y5 + y6 - #6.0) - #0.151 * (y1 + y2 + y3 - #6.0) > #0.0)`; };; add{ idv = "4047599236"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3 y1 y2 y3 y4 y5 y6) ( (dih_y y1 y2 y3 y4 y5 y6) - #1.2308 + (#0.3639 * (y2 + y3 + y5 + y6 - #8.0)) - (#0.235 * (y1 - #2.0)) -(#0.685 * (y4 - #2.0)) > #0.0)`; };; add{ idv = "3526497018"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3 y1 y2 y3 y4 y5 y6) ( (--dih_y y1 y2 y3 y4 y5 y6) + #1.231 - (#0.152 * (y2 + y3 + y5 + y6 - #8.0))+ (#0.5 * (y1 - #2.0)) + (#0.773 * (y4 - #2.0))> #0.0)`; };; add{ idv = "5957966880"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3 y1 y2 y3 y4 y5 y6) ( (rhazim y1 y2 y3 y4 y5 y6) - #1.2308 + (#0.3639 * (y2 + y3 + y5 + y6 - #8.0)) - (#0.6 * (y1 - #2.0)) -(#0.685 * (y4 - #2.0)) > #0.0)`; };; addtex(Section,"Linear Programs"," -- dartXYZ");; (*more interval arithmetic on nonstandard triangles*) let dartX = define_dart `dartX y1 y2 y3 y4 y5 y6 = [ (#2.0,y1,#2.52); (#2.0,y2,#2.52); (#2.0,y3,#2.52); (#2.52,y4,#2.52); (#2.0,y5,#2.52); (#2.0,y6,#2.52)]`;; add{ idv = "3020140039"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dartX y1 y2 y3 y4 y5 y6) ( (dih_y y1 y2 y3 y4 y5 y6) - #1.629 + (#0.402 * (y2 + y3 + y5 + y6 - #8.0)) - (#0.315 * (y1 - #2.0)) > #0.0)`; };; let dartY = define_dart `dartY y1 y2 y3 y4 y5 y6 = [ (#2.0,y1,#2.52); (#2.0,y2,#2.52); (#2.0,y3,#2.52); (sqrt8,y4,sqrt8); (#2.0,y5,#2.52); (#2.0,y6,#2.52)]`;; add{ idv = "9414951439"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dartY y1 y2 y3 y4 y5 y6) ( (dih_y y1 y2 y3 y4 y5 y6) - #1.91 + (#0.458 * (y2 + y3 + y5 + y6 - #8.0)) - (#0.342 * (y1 - #2.0)) > #0.0)` ; };; let dart4_diag3 = define_dart `dart4_diag3 y1 y2 y3 y4 y5 y6 = [ (#2.0,y1,#2.52); (#2.0,y2,#2.52); (#2.0,y3,#2.52); (#3.0,y4,#3.0); (#2.0,y5,#2.52); (#2.0,y6,#2.52)]`;; add{ idv = "9995621667"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart4_diag3 y1 y2 y3 y4 y5 y6) ( (dih_y y1 y2 y3 y4 y5 y6) - #2.09 + (#0.578 * (y2 + y3 + y5 + y6 - #8.0)) - (#0.54 * (y1 - #2.0)) > #0.0)`; };; (*branch flat inequality*) let apex_flat = define_dart `apex_flat y1 y2 y3 y4 y5 y6 = [ (#2.0,y1,#2.52); (#2.0,y2,#2.52); (#2.0,y3,#2.52); (#2.52,y4,sqrt8); (#2.0,y5,#2.52); (#2.0,y6,#2.52)]`;; add{ idv = "6988401556"; doc="We can use extremal edge properties and the tame table D calculations to reduce to the case where $y_4=\\sqrt8$ and $y_5,y_6$ are extremal."; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (apex_flat y1 y2 y3 y4 y5 y6) (taum y1 y2 y3 y4 y5 y6 > #0.103)`; };; add{ idv = "8248508703"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (apex_flat y1 y2 y3 y4 y5 y6) ( (taum y1 y2 y3 y4 y5 y6) - #0.1 - (#0.265 * (y5 + y6 - #4.0)) - (#0.06 * (y4 - #2.52)) - (#0.16 * (y1 - #2.0)) - (#0.115 * (y2 + y3 - #4.0)) > #0.0)`; };; add{ idv = "3318775219"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (apex_flat y1 y2 y3 y4 y5 y6) ( (dih_y y1 y2 y3 y4 y5 y6) - #1.629 + (#0.414 * (y2 + y3 + y5 + y6 - #8.0)) - (#0.763 * (y4 - #2.52)) - (#0.315 * (y1 - #2.0)) > #0.0)`; };; add{ idv = "9922699028"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (apex_flat y1 y2 y3 y4 y5 y6) ( (--dih_y y1 y2 y3 y4 y5 y6) + #1.6294 - (#0.2213 * (y2 + y3 + y5 + y6 - #8.0)) + (#0.913 * (y4 - #2.52)) + (#0.728 * (y1 - #2.0)) > #0.0)`; };; add{ idv = "5000076558"; doc=""; tags = [Cfsqp;Xconvert;Lp;Lpsymmetry;Tex]; ineq = all_forall `ineq (apex_flat y1 y2 y3 y4 y5 y6) ( (dih2_y y1 y2 y3 y4 y5 y6) - #1.083 + (#0.6365 * (y1 - #2.0)) - (#0.198 * (y2 - #2.0)) + (#0.352 * (y3 - #2.0)) + (#0.416 * (y4 - #2.52)) - (#0.66 * (y5 - #2.0)) + (#0.071 * (y6 - #2.0)) > #0.0)`; };; add{ idv = "9251360200"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (apex_flat y1 y2 y3 y4 y5 y6) ( (rhazim y1 y2 y3 y4 y5 y6) - #1.629 - (#0.866 * (y1 - #2.0)) + (#0.3805 * (y2 + y3 - #4.0)) - (#0.841 * (y4 - #2.52)) + (#0.501 * (y5 + y6 - #4.0)) > #0.0)`; };; add{ idv = "9756015945"; doc=""; tags = [Cfsqp;Xconvert;Lp;Lpsymmetry;Tex]; ineq = all_forall `ineq (apex_flat y1 y2 y3 y4 y5 y6) ( (rhazim2 y1 y2 y3 y4 y5 y6) - #1.08 + (#0.6362 * (y1 - #2.0)) - (#0.565 * (y2 - #2.0)) + (#0.359 * (y3 - #2.0)) + (#0.416 * (y4 - #2.52)) - (#0.666 * (y5 - #2.0)) + (#0.061 * (y6 - #2.0)) > #0.0)`; };; let apex_A = define_dart `apex_A y1 y2 y3 y4 y5 y6 = [(#2.0, y1, #2.52); (#2.0, y2, #2.52); (#2.0, y3, #2.52); (#2.0, y4, #2.52); (#2.52, y5, sqrt8); (#2.52, y6, sqrt8)]`;; add{ idv = "8082208587"; doc="We can use extremal edge properties and the tame table D (1,2) calculations to reduce to the case where $y_4=\\sqrt8$ and $y_5,y_6$ are extremal."; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (apex_A y1 y2 y3 y4 y5 y6) (taum y1 y2 y3 y4 y5 y6 > #0.2759)`; };; add{ idv = "5760733457"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (apex_A y1 y2 y3 y4 y5 y6) (dih_y y1 y2 y3 y4 y5 y6 - #1.0705 -( #0.1 * (y1 - #2)) + (#0.424 * (y2 - #2.0)) + (#0.424 * (y3 - #2.0)) - (#0.594 * (y4 - #2.0)) + (#0.124 * (y5 - #2.52)) + (#0.124 * (y6 - #2.52)) > #0.0)`; };; add{ idv = "2563100177"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (apex_A y1 y2 y3 y4 y5 y6) ( rhazim y1 y2 y3 y4 y5 y6 - #1.0685 - (#0.4635 * (y1 - #2.0)) + (#0.424 * (y2 - #2.0)) + (#0.424 * (y3 - #2.0)) - (#0.594 * (y4 - #2.0)) + (#0.124 * (y5 - #2.52)) + (#0.124 * (y6 - #2.52)) > #0.0)`; };; add{ idv = "7931207804"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (apex_A y1 y2 y3 y4 y5 y6) ( taum y1 y2 y3 y4 y5 y6 - #0.27 + (#0.0295 * (y1 - #2.0)) - (#0.0778 * (y2 - #2.0)) - (#0.0778 * (y3 - #2.0)) - (#0.37 * (y4 - #2.0)) - (#0.27 * (y5 - #2.52)) - (#0.27 * (y6 - #2.52)) > #0.0)`; };; let dart_std3_small = define_dart `dart_std3_small y1 y2 y3 y4 y5 y6 = [(#2.0, y1, #2.52); (#2.0, y2, #2.52); (#2.0, y3, #2.52); (#2.0, y4, #2.25); (#2.0, y5, #2.25); (#2.0, y6, #2.25)]`;; add{ idv = "9225295803"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3_small y1 y2 y3 y4 y5 y6) (taum y1 y2 y3 y4 y5 y6 + #0.0034 - (#0.166 * (y1 + y2 + y3 - #6.0)) - (#0.22 * (y4 + y5 + y6 - #6.0)) > #0.0)`; };; add{ idv = "9291937879"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3_small y1 y2 y3 y4 y5 y6) (dih_y y1 y2 y3 y4 y5 y6 - #1.23 - (#0.235 * (y1 - #2.0)) + (#0.362 * (y2 + y3 - #4.0)) - (#0.694 * (y4 - #2.0)) + (#0.26 * (y5 + y6 - #4.0)) > #0.0)`; };; (* same domain but extra disjunct *) let dart_std3_big = define_dart `dart_std3_big = dart_std3`;; add{ idv = "7761782916"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3_big y1 y2 y3 y4 y5 y6) ((taum y1 y2 y3 y4 y5 y6 - #0.05 - (#0.137 * (y1 + y2 + y3 - #6.0)) - (#0.17 * (y4 + y5 + y6 - #6.25)) > #0.0)\/ (y4 + y5 + y6 < #6.25))`; };; add{ idv = "6224332984"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3_big y1 y2 y3 y4 y5 y6) ((sol_y y1 y2 y3 y4 y5 y6 - #0.589 + (#0.39 * (y1 + y2 + y3 - #6.0)) - (#0.235 * (y4 + y5 + y6 - #6.25)) > #0.0) \/ (y4 + y5 + y6 < #6.25))`; };; let apex_sup_flat = define_dart `apex_sup_flat y1 y2 y3 y4 y5 y6 = [(#2.0, y1, #2.52); (#2.0, y2, #2.52); (#2.0, y3, #2.52); (sqrt8, y4, #3.0); (#2.0, y5, #2.52); (#2.0, y6, #2.52)] `;; add{ idv = "5451229371"; doc=""; tags = [Cfsqp;Tex;Xconvert;Lp]; ineq = all_forall `ineq (apex_sup_flat y1 y2 y3 y4 y5 y6) (taum y1 y2 y3 y4 y5 y6 - #0.11 - #0.14132 * (y1 + (y2 + y3) / &2 - &4) - #0.38 * (y5 + y6 - &4) > &0)`; };; add{ idv = "4840774900"; doc=""; tags = [Cfsqp;Tex;Xconvert;Lp]; ineq = all_forall `ineq (apex_sup_flat y1 y2 y3 y4 y5 y6) ( taum y1 y2 y3 y4 y5 y6 - #0.1054 - #0.14132*(y1 + y2 / &2 + y3 / &2 - &4) - #0.36499*(y5 +y6 - &4) > &0)`; };; add{ idv = "1642527039"; doc=""; tags = [Cfsqp;Tex;Xconvert;Lp]; ineq = all_forall `ineq (apex_sup_flat y1 y2 y3 y4 y5 y6) ( taum y1 y2 y3 y4 y5 y6 - #0.128 - #0.053*((y5 +y6 - &4) - (#2.75/ &2)*(y4 - sqrt8)) > &0)`; };; add{ idv = "7863247282"; doc=""; tags = [Cfsqp;Tex;Xconvert;Lp]; ineq = all_forall `ineq (apex_sup_flat y1 y2 y3 y4 y5 y6) (taum y1 y2 y3 y4 y5 y6 - #0.053*((y5 +y6 - &4) - (#2.75/ &2)*(y4 - sqrt8)) - #0.12 - #0.14132*(y1 + y2 / &2 + y3 / &2 - &4) - #0.328*(y5 +y6 - &4) > &0)`; (* corrected 2010-06-22, thales, was y4 + y5, was >= *) };; add{ idv = "7718591733"; doc=""; tags = [Cfsqp;Tex;Xconvert;Lp]; ineq = all_forall `ineq (apex_sup_flat y1 y2 y3 y4 y5 y6) (dih2_y y1 y2 y3 y4 y5 y6 - #0.955 - #0.2356*(y2 - &2) + #0.32*(y3 - &2) + #0.792*(y1 - &2) - #0.707*(y5 - &2) + #0.0844*(y6 - &2) + #0.821*(y4 - sqrt8) > &0)`; (* corrected 2010-06-22, was >= *) };; add{ idv = "3566713650"; doc=""; tags = [Cfsqp;Tex;Xconvert;Lp]; ineq = all_forall `ineq (apex_sup_flat y1 y2 y3 y4 y5 y6) ( -- dih_y y1 y2 y3 y4 y5 y6 + #1.911 + #1.01 *(y1 - &2) - #0.284*(y2 +y3 +y5 +y6 - &8) + #1.07*(y4 - sqrt8) > &0)`; (* was >= *) };; add{ idv = "1085358243"; doc=""; tags = [Cfsqp;Tex;Xconvert;Lp]; ineq = all_forall `ineq (apex_sup_flat y1 y2 y3 y4 y5 y6) ( dih_y y1 y2 y3 y4 y5 y6 - #1.903 - #0.4*(y1 - &2) + #0.49688*(y2 +y3 +y5 +y6 - &8) -(y4 -sqrt8) > &0)`; };; (* use this if there is no smallness constraint *) let dart_std3_mini = define_dart `dart_std3_mini y1 y2 y3 y4 y5 y6 = [ (#2,y1,#2.18); (&2,y2,#2.18); (&2,y3,#2.18); (&2,y4,#2.25); (&2,y5,#2.25); (&2,y6,#2.25) ]` ;; add{ idv = "9229542852"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3_mini y1 y2 y3 y4 y5 y6) (dih_y y1 y2 y3 y4 y5 y6 - #1.230 - (#0.2357 * (y1 - #2.0)) + (#0.2493 * (y2 + y3 - #4.0)) - (#0.682 * (y4 - #2.0)) + (#0.3035 * (y5 + y6 - #4.0)) > #0.0)`; };; add{ idv = "1550635295"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3_mini y1 y2 y3 y4 y5 y6) (--(dih_y y1 y2 y3 y4 y5 y6) + #1.232 + (#0.261 * (y1 - #2.0)) - (#0.203 * (y2 + y3 - #4.0)) + (#0.772 * (y4 - #2.0)) - (#0.191 * (y5 + y6 - #4.0)) > #0.0)`; };; add{ idv = "4491491732"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3_mini y1 y2 y3 y4 y5 y6) (taum y1 y2 y3 y4 y5 y6 + #0.0008 - (#0.1631 * (y1 + y2 + y3 - #6.0)) - (#0.2127 * (y4 + y5 + y6 - #6.0)) > #0.0)`; };; let apex_flat_hll = define_dart `apex_flat_hll y1 y2 y3 y4 y5 y6 = [(#2.18, y1, #2.52); (#2.0, y2, #2.18); (#2.0, y3, #2.18); (#2.52, y4, sqrt8); (#2.0, y5, #2.52); (#2.0, y6, #2.52)]`;; add{ idv = "8282573160"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (apex_flat_hll y1 y2 y3 y4 y5 y6) (taum y1 y2 y3 y4 y5 y6 - #0.1413 - (#0.214 * (y1 - #2.18)) - (#0.1259 * (y2 + y3 - #4.0)) - (#0.067 * (y4 - #2.52)) - (#0.241 * (y5 + y6 - #4.0)) > #0.0)`; };; let dart_std3_big_200_218 = define_dart `dart_std3_big_200_218 y1 y2 y3 y4 y5 y6 = [(#2.0, y1, #2.18); (#2.0, y2, #2.18); (#2.0, y3, #2.18); (#2.0, y4, #2.52); (#2.0, y5, #2.52); (#2.0, y6, #2.52)]`;; add{ idv = "8611785756"; doc=""; tags = [Cfsqp;Xconvert;Lp;Tex]; ineq = all_forall `ineq (dart_std3_big_200_218 y1 y2 y3 y4 y5 y6) ((sol_y y1 y2 y3 y4 y5 y6 - #0.589 + (#0.24 * (y1 + y2 + y3 - #6.0)) - (#0.16 * (y4 + y5 + y6 - #6.25)) > #0.0) \/ (y4 + y5 + y6 < #6.25))`; };; (* 181212899 is applied in six different ways, each with a different dart class. *) let dart_template = `\dart_class y4min y4max. dart_class (y1:real) (y2:real) (y3:real) (y4:real) (y5:real) (y6:real) = [ (#2.0, y1, #2.52); (#2.0, y2, #2.52); (#2.0, y3, #2.52); (y4min, y4, y4max); (#2.0, y5, #2.52); (#2.52, y6, sqrt8)]`;; let dart_template_reverse = `\dart_class y4min y4max. dart_class (y1:real) (y2:real) (y3:real) (y4:real) (y5:real) (y6:real) = [ (#2.0, y1, #2.52); (#2.0, y2, #2.52); (#2.0, y3, #2.52); (y4min, y4, y4max); (#2.52, y5, sqrt8); (#2.0, y6, #2.52)]`;; let tyR = `:real->real->real->real->real->real-> (real#real#real)list`;; let mk_dart rev (x,y,z) = let x' = if (is_var x) then mk_var(fst(dest_var x),tyR) else x in let plate = if (rev) then dart_template_reverse else dart_template in snd(strip_forall (mk_tplate plate [x';y;z]));; let newdef rev t = define_dart (mk_dart rev t);; let apexffA = newdef false (`apexffA`,`#2.52`,`sqrt8`);; (* NB ff comes before f on this *) let apexfA = newdef true (`apexfA`,`#2.52`,`sqrt8`);; let apexf4 = newdef false (`apexf4`,`sqrt8`,`sqrt8`);; let apexff4 = newdef true (`apexff4`,`sqrt8`,`sqrt8`);; let apexf5 = newdef false (`apexf5`,`#2.52`,`#2.52`);; let apexff5 = newdef true (`apexff5`,`#2.52`,`#2.52`);; let template_181212899= `\dart_class y4e y2' y3' y5' y6'. ineq (dart_class y1 y2 y3 y4 y5 y6) (dih_y y1 y2 y3 y4 y5 y6 - #1.448 - (#0.266 * (y1 - #2.0)) + (#0.295 * (y2' - #2.0)) + (#0.57 * (y3' - #2.0)) - (#0.745 * (y4e - #2.52)) + (#0.268 * (y5' - #2.0)) + (#0.385 * (y6' - #2.52)) > #0.0)`;; (* coefficients corrected, April 28, 2010, June 20, 2010-thales *) let mk_iqd rev (d,dart,y4e) = let yy = if rev then [`y3:real`;`y2:real`;`y6:real`;`y5:real`] else [`y2:real`;`y3:real`;`y5:real`;`y6:real`] in let ineq = mk_tplate template_181212899([dart;y4e] @ yy) in { idv = Printf.sprintf "181212899 %d" d; doc = if (d=0) then "This original version of the inequality. There are five other derived versions." else ""; tags = [Cfsqp;Xconvert;Lp] @ (if (d>0) then [Derived []] else [Tex]); ineq = all_forall ineq; };; add (mk_iqd false (0,`apexffA`,`y4:real`));; add (mk_iqd true (1,`apexfA`,`y4:real`));; add (mk_iqd false (2,`apexf4`,`sqrt8`));; add (mk_iqd true (3,`apexff4`,`sqrt8`));; add (mk_iqd false (4,`apexf5`,`#2.52`));; add (mk_iqd true (5,`apexff5`,`#2.52`));; addtex(Section,"Linear Programs"," -- apex std3 hll");; let apex_std3_hll = define_dart `apex_std3_hll y1 y2 y3 y4 y5 y6 = [ (#2.18,y1,#2.52); (&2,y2,#2.18); (&2,y3,#2.18); (&2,y4,#2.52); (&2,y5,#2.52); (&2,y6,#2.52) ]`;; let apex_std3_small_hll = define_dart `apex_std3_small_hll y1 y2 y3 y4 y5 y6 = [ (#2.18,y1,#2.52); (&2,y2,#2.18); (&2,y3,#2.18); (&2,y4,#2.25); (&2,y5,#2.25); (&2,y6,#2.25) ]`;; let dart_mll_w = define_dart `dart_mll_w y1 y2 y3 y4 y5 y6 = [ (#2.18,y1,#2.36); (&2,y2,#2.18); (&2,y3,#2.18); (#2.25,y4,#2.52); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; let dart_mll_n = define_dart `dart_mll_n y1 y2 y3 y4 y5 y6 = [ (#2.18,y1,#2.36); (&2,y2,#2.18); (&2,y3,#2.18); (&2,y4,#2.25); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; let dart_Hll_n = define_dart `dart_Hll_n y1 y2 y3 y4 y5 y6 = [ (#2.36,y1,#2.52); (&2,y2,#2.18); (&2,y3,#2.18); (&2,y4,#2.25); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; let dart_Hll_w = define_dart `dart_Hll_w y1 y2 y3 y4 y5 y6 = [ (#2.36,y1,#2.52); (&2,y2,#2.18); (&2,y3,#2.18); (#2.25,y4,#2.52); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; let dart_mll_w = define_dart `dart_mll_w y1 y2 y3 y4 y5 y6 = [ (#2.18,y1,#2.36); (&2,y2,#2.18); (&2,y3,#2.18); (#2.25,y4,#2.52); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; (* repeated *) let dart_mll_n = define_dart `dart_mll_n y1 y2 y3 y4 y5 y6 = [ (#2.18,y1,#2.36); (&2,y2,#2.18); (&2,y3,#2.18); (&2,y4,#2.25); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; let dart_Hll_n = define_dart `dart_Hll_n y1 y2 y3 y4 y5 y6 = [ (#2.36,y1,#2.52); (&2,y2,#2.18); (&2,y3,#2.18); (&2,y4,#2.25); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; let dart_Hll_w = define_dart `dart_Hll_w y1 y2 y3 y4 y5 y6 = [ (#2.36,y1,#2.52); (&2,y2,#2.18); (&2,y3,#2.18); (#2.25,y4,#2.52); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; let templateC = `\ dart_class f fc f0 c1 c2 c3 c4 c5 c6 y1min y2min y3min y4min y5min y6min . ineq (dart_class y1 y2 y3 y4 y5 y6) ( fc * f y1 y2 y3 y4 y5 y6 > f0 + c1 * (y1 - y1min) + c2 *(y2- y2min) + c3 * (y3 - y3min) + c4 * (y4 - y4min) + c5 *(y5- y5min) + c6 * (y6 - y6min))`;; let yymin = dest_list `[#2.18;&2;&2;&2;&2;&2]`;; let iqd s dart sym t = let c = mk_mconst (dart,tyR) in { idv = s; doc = ""; tags = [Cfsqp;Xconvert;Lp;] @ (if sym then [Lpsymmetry] else []); ineq = mk_tplate templateC (c:: t); };; add(iqd "2151506422" "apex_std3_hll" false ([`dih_y`;`&1`;`#1.2777`] @ dest_list `[#0.281; -- #0.278364; -- #0.278364; #0.7117; -- #0.34336; -- #0.34336]` @ yymin));; add(iqd "6836427086" "apex_std3_hll" false ([`dih_y`;`-- &1`;`-- #1.27799`] @ dest_list `[-- #0.356217; #0.229466; #0.229466; -- #0.949067; #0.172726; #0.172726]` @ yymin));; add(iqd "3636849632" "apex_std3_hll" false ([`taum`;`&1`;`#0.0345`] @ dest_list `[#0.185545; #0.193139; #0.193139; #0.170148; #0.13195; #0.13195]` @ yymin));; add(iqd "5298513205" "apex_std3_hll" true ([`dih2_y`;`&1`;`#1.185`] @ dest_list `[-- #0.302913; #0.214849; -- #0.163775; -- #0.443449; #0.67364; -- #0.314532]` @ yymin));; add(iqd "7743522046" "apex_std3_hll" true ([`dih2_y`;`-- &1`;`-- #1.1865`] @ dest_list `[#0.20758; -- #0.236153; #0.14172; #0.263834; -- #0.771203; #0.0457292]` @ yymin));; let yymin_plus = dest_list `[#2.36;&2;&2;#2.25;&2;&2]`;; add(iqd "8657368829" "apex_std3_small_hll" false ([`dih_y`;`&1`;`#1.277`] @ dest_list `[#0.273298; -- #0.273853; -- #0.273853; #0.708818; -- #0.313988; -- #0.313988]` @ yymin));; add(iqd "6619134733" "apex_std3_small_hll" false ([`dih_y`;`-- &1`;`-- #1.27799`] @ dest_list `[-- #0.439002; #0.229466; #0.229466; -- #0.771733; #0.208429; #0.208429]` @ yymin));; add(iqd "1284543870" "apex_std3_small_hll" true ([`dih2_y`;`&1`;`#1.185`] @ dest_list `[-- #0.372262; #0.214849; -- #0.163775; -- #0.293508; #0.656172; -- #0.267157]` @ yymin));; add(iqd "4041673283" "apex_std3_small_hll" true ([`dih2_y`;`-- &1`;`-- #1.1864`] @ dest_list `[#0.20758; -- #0.236153; #0.14172; #0.263109; -- #0.737003; #0.12047]` @ yymin));; add(iqd "3872614111" "dart_mll_w" false ([`dih_y`;`-- &1`;`-- #1.542`] @ dest_list `[-- #0.362519; #0.298691; #0.287065; -- #0.920785; #0.190917; #0.219132]` @ yymin_plus));; add(iqd "3139693500" "dart_mll_n" false ([`dih_y`;`-- &1`;`-- #1.542`] @ dest_list `[-- #0.346773; #0.300751; #0.300751; -- #0.702567; #0.172726; #0.172727]` @ yymin_plus));; add(iqd "4841020453" "dart_Hll_n" false ([`dih_y`;`-- &1`;`-- #1.542`] @ dest_list `[-- #0.490439; #0.318125; #0.32468; -- #0.740079; #0.178868; #0.205819]` @ yymin_plus));; add(iqd "9925287433" "dart_Hll_w" false ([`dih_y`;`-- &1`;`-- #1.542`] @ dest_list `[-- #0.490439; #0.321849; #0.320956; -- #1.00902; #0.240709; #0.218081]` @ yymin_plus));; add(iqd "7409690040" "dart_mll_w" true ([`dih2_y`;`&1`;`#1.0494`] @ dest_list `[-- #0.297823; #0.215328; -- #0.0792439; -- #0.422674; #0.647416; -- #0.207561]` @ yymin_plus));; add(iqd "4002562507" "dart_mll_n" true ([`dih2_y`;`&1`;`#1.0494`] @ dest_list `[-- #0.29013; #0.215328; -- #0.0715511; -- #0.267157; #0.650269; -- #0.295198]` @ yymin_plus));; add(iqd "5835568093" "dart_Hll_n" true ([`dih2_y`;`&1`;`#1.0494`] @ dest_list `[-- #0.404131; #0.212119; -- #0.0402827; -- #0.299046; #0.643273; -- #0.266118]` @ yymin_plus));; add(iqd "1894886027" "dart_Hll_w" true ([`dih2_y`;`&1`;`#1.0494`] @ dest_list `[-- #0.401543; #0.207551; -- #0.0294227; -- #0.494954; #0.605453; -- #0.156385]` @ yymin_plus));; (* NOW DATA SECTION OF NEW DART CLASSES THAT HAVE BEEN FOUND *) let dart_std3_lw = define_dart `dart_std3_lw y1 y2 y3 y4 y5 y6 = [ (#2.00,y1,#2.18); (&2,y2,#2.52); (&2,y3,#2.52); (#2.25,y4,#2.52); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; (* NOW DATA SECTION OF NEW INEQUALITIES THAT HAVE BEEN FOUND *) (* generate_ineq_datum "mDihedral2" "{2,2,2,2.52,2,2}" "{2.52,2.52,2.52,sqrt8,2.52,2.52}";; Annoyance: have to manually edit `--#X.XX` -> `-- #X.XX` *) let i4750199435 = {ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_flat y1 y2 y3 y4 y5 y6) (--dih2_y y1 y2 y3 y4 y5 y6 + #0.0031 > -- #1.08346 + #0.288794 * (-- &2 + y1) + -- #0.292829 * (-- &2 + y2) + #0.036457 * (-- &2 + y3) + #0.348796 * (-- #2.52 + y4) + -- #0.762602 * (-- &2 + y5) + -- #0.112679 * (-- &2 + y6))`; idv = "4750199435"; doc = "date{2010, 8, 1, 7, 59, 42.645902} ineq generated by Mathematica holineq The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add i4750199435;; (* 8384511215;; generate_ineq_datum_p "Dihedral2" "{2,2,2.52,sqrt8,2,2}" "{2,2,2,2.52,2,2}" "{2.52,2.52,2.52,sqrt8,2.52,2.52}";; *) let i8384511215 = {ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_flat y1 y2 y3 y4 y5 y6) (dih2_y y1 y2 y3 y4 y5 y6 + #0.0015 > #0.913186 + -- #0.390288 * (-- &2 + y1) + #0.115895 * (-- &2 + y2) + #0.164805 * (-- #2.52 + y3) + -- #0.271329 * (-- #2.82843 + y4) + #0.584817 * (-- &2 + y5) + -- #0.170218 * (-- &2 + y6))`; idv = "8384511215"; doc = "date{2010, 8, 1, 12, 24, 31.880776}\n ineq generated by Mathematica holineq\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add i8384511215;; (* 7819193535; generate_ineq_datum "Dihedral2" "{2,2,2,2.25,2,2}" "{2.18,2.52,2.52,2.52,2.52,2.52}";; *) let i7819193535 = {ineq = `!y1 y2 y3 y4 y5 y6. ineq (dart_std3_lw y1 y2 y3 y4 y5 y6) (dih2_y y1 y2 y3 y4 y5 y6 + #0.0011 > #1.16613 + -- #0.296776 * (-- &2 + y1) + #0.208935 * (-- &2 + y2) + -- #0.243302 * (-- &2 + y3) + -- #0.360575 * (-- #2.25 + y4) + #0.636205 * (-- &2 + y5) + -- #0.295156 * (-- &2 + y6))`; idv = "7819193535"; doc = "date{2010, 8, 1, 14, 6, 32.558867}\n ineq generated by Mathematica holineq\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add i7819193535;; (* remove "2621779878";; *) (* let ii = generate_ineq_datum_p "Dihedral2" "{2.36,2,2,2.52,2,2.25}" "{2.18,2,2,2.25,2,2}" "{2.36,2.18,2.18,2.52,2.52,2.52}";; *) let i6987934000 = {ineq = `!y1 y2 y3 y4 y5 y6. ineq (dart_mll_w y1 y2 y3 y4 y5 y6) (dih2_y y1 y2 y3 y4 y5 y6 + #0.0042 > #0.952682 + -- #0.268837 * (-- #2.36 + y1) + #0.130607 * (-- &2 + y2) + -- #0.168729 * (-- &2 + y3) + -- #0.0831764 * (-- #2.52 + y4) + #0.580152 * (-- &2 + y5) + #0.0656612 * (-- #2.25 + y6))`; idv = "6987934000"; doc = "date{2010, 8, 1, 17, 50, 0.245131}\n ineq generated by Mathematica holineq\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add i6987934000;; let i7291663656 = {ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_flat y1 y2 y3 y4 y5 y6) (dih2_y y1 y2 y3 y4 y5 y6 + #0.0009 > #0.947391 + -- #0.637397 * (-- &2 + y1) + #0.120003 * (-- &2 + y2) + -- #0.100814 * (-- #2.3 + y3) + -- #0.302956 * (-- #2.65 + y4) + #0.547359 * (-- &2 + y5) + -- #0.157745 * (-- #2.2 + y6))`; idv = "7291663656"; doc = "date{2010, 8, 2, 15, 44, 43.580639}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input Dihedral2 {2,2,2.3,2.65,2,2.2} {2,2,2,2.52,2,2} {2.52,2.52,2.52,sqrt8,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add i7291663656;; let i2390583444= {ineq = `!y1 y2 y3 y4 y5 y6. ineq (dart_std3_mini y1 y2 y3 y4 y5 y6) (dih_y y1 y2 y3 y4 y5 y6 + #0.0012 > #1.08627 + #0.159149 * (-- &2 + y1) + -- #0.198496 * (-- #2.1 + y2) + -- #0.199306 * (-- #2.1 + y3) + #0.590083 * (-- &2 + y4) + -- #0.0888111 * (-- #2.25 + y5) + -- #0.0881846 * (-- #2.25 + y6))`; idv = "2390583444"; doc = "date{2010, 8, 2, 17, 51, 29.665035}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input Dihedral {2,2.1,2.1,2,2.25,2.25} {2,2,2,2,2,2} {2.18,2.18,2.18,2.25,2.25,2.25}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add i2390583444;; let apex_flat_l = define_dart `apex_flat_l y1 y2 y3 y4 y5 y6 = [ (#2,y1,#2.18); (&2,y2,#2.52); (&2,y3,#2.52); (#2.52,y4,sqrt8); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; let apex_flat_h = define_dart `apex_flat_h y1 y2 y3 y4 y5 y6 = [ (#2.18,y1,#2.52); (&2,y2,#2.52); (&2,y3,#2.52); (#2.52,y4,sqrt8); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_flat_l y1 y2 y3 y4 y5 y6) (dih2_y y1 y2 y3 y4 y5 y6 + #0.0071 > #0.98362 + -- #0.264094 * (-- #2.18 + y1) + #0.149308 * (-- #2.18 + y2) + -- #0.312683 * (-- &2 + y3) + -- #0.282792 * (-- #2.65 + y4) + #0.581552 * (-- &2 + y5) + #0.143669 * (-- #2.3 + y6))`; idv = "9641946727"; doc = "date{2010, 8, 2, 20, 0, 47.561405}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input Dihedral2 {2.18,2.18,2,2.65,2,2.3} {2,2,2,2.52,2,2} {2.18,2.52,2.52,sqrt8,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; let apex_std3_lll_xww = define_dart `apex_std3_lll_xww y1 y2 y3 y4 y5 y6 = [ (&2,y1,#2.18); (&2,y2,#2.18); (&2,y3,#2.18); (&2,y4,#2.52); (#2.25,y5,#2.52); (#2.25,y6,#2.52) ]` ;; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_std3_lll_xww y1 y2 y3 y4 y5 y6) (dih_y y1 y2 y3 y4 y5 y6 + #0.0071 > #1.09969 + #0.146345 * (-- &2 + y1) + -- #0.160538 * (-- &2 + y2) + -- #0.151698 * (-- #2.14 + y3) + #0.61314 * (-- &2 + y4) + -- #0.236149 * (-- #2.25 + y5) + -- #0.242043 * (-- #2.25 + y6))`; idv = "4222324842"; doc = "date{2010, 8, 3, 7, 9, 1.995901}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input Dihedral {2,2,2.14,2,2.25,2.25} {2,2,2,2,2.25,2.25} {2.18,2.18,2.18,2.52,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; let apex_std3_lll_wxx = define_dart `apex_std3_lll_wxx y1 y2 y3 y4 y5 y6 = [ (&2,y1,#2.18); (&2,y2,#2.18); (&2,y3,#2.18); (#2.25,y4,#2.52); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_std3_lll_wxx y1 y2 y3 y4 y5 y6) (dih2_y y1 y2 y3 y4 y5 y6 + #0.0025 > #1.16613 + -- #0.296776 * (-- &2 + y1) + #0.208935 * (-- &2 + y2) + -- #0.196313 * (-- &2 + y3) + -- #0.360575 * (-- #2.25 + y4) + #0.652861 * (-- &2 + y5) + -- #0.218063 * (-- &2 + y6))`; idv = "5756588587"; doc = "date{2010, 8, 3, 19, 47, 22.770311}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum with input Dihedral2 {2,2,2,2.25,2,2} {2.18,2.18,2.18,2.52,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_flat y1 y2 y3 y4 y5 y6) (--dih_y y1 y2 y3 y4 y5 y6 + #0.0011 > -- #1.67609 + -- #0.506322 * (-- #2.18 + y1) + #0.212075 * (-- #2.1 + y2) + #0.230669 * (-- #2.1 + y3) + -- #1.28579 * (-- #2.52 + y4) + #0.249199 * (-- &2 + y5) + #0.193545 * (-- &2 + y6))`; idv = "3425739813"; doc = "date{2010, 8, 3, 20, 2, 51.133190}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input mDihedral {2.18,2.1,2.1,2.52,2,2} {2,2,2,2.52,2,2} {2.52,2.52,2.52,sqrt8,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_std3_lll_wxx y1 y2 y3 y4 y5 y6) (dih2_y y1 y2 y3 y4 y5 y6 + #0.0050 > #1.02005 + -- #0.256494 * (-- #2.18 + y1) + #0.121497 * (-- &2 + y2) + -- #0.256494 * (-- &2 + y3) + -- #0.0116869 * (-- #2.52 + y4) + #0.598233 * (-- &2 + y5) + #0.0187672 * (-- #2.25 + y6))`; idv = "7316455966"; doc = "date{2010, 8, 4, 6, 2, 11.667622}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input Dihedral2 {2.18,2,2,2.52,2,2.25} {2,2,2,2.25,2,2} {2.18,2.18,2.18,2.52,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_std3_lll_wxx y1 y2 y3 y4 y5 y6) (--dih3_y y1 y2 y3 y4 y5 y6 + #0.0087 > -- #1.18616 + #0.436647 * (-- #2.18 + y1) + #0.032258 * (-- &2 + y2) + -- #0.289629 * (-- &2 + y3) + #0.397053 * (-- #2.52 + y4) + -- #0.0210289 * (-- &2 + y5) + -- #0.683341 * (-- #2.25 + y6))`; idv = "6410081357"; doc = "date{2010, 8, 4, 6, 2, 43.302104}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input mDihedral3 {2.18,2,2,2.52,2,2.25} {2,2,2,2.25,2,2} {2.18,2.18,2.18,2.52,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (dart_mll_n y1 y2 y3 y4 y5 y6) (--dih_y y1 y2 y3 y4 y5 y6 + #0.0083 > -- #1.29912 + -- #0.284457 * (-- #2.18 + y1) + #0.337354 * (-- &2 + y2) + #0.186287 * (-- &2 + y3) + -- #0.645382 * (-- #2.25 + y4) + #0.367671 * (-- #2.52 + y5) + #0.0536051 * (-- &2 + y6))`; idv = "2923748598"; doc = "date{2010, 8, 4, 8, 57, 40.892442}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input mDihedral {2.18,2,2,2.25,2.52,2} {2.18,2,2,2,2,2} {2.36,2.18,2.18,2.25,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (dart_mll_n y1 y2 y3 y4 y5 y6) (dih2_y y1 y2 y3 y4 y5 y6 + #0.0035 > #1.05036 + -- #0.222178 * (-- #2.18 + y1) + #0.132628 * (-- &2 + y2) + -- #0.219284 * (-- &2 + y3) + #0.00563427 * (-- #2.25 + y4) + #0.59096 * (-- &2 + y5) + -- #0.0771574 * (-- #2.52 + y6))`; idv = "4306175952"; doc = "date{2010, 8, 4, 9, 3, 43.896393}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input Dihedral2 {2.18,2,2,2.25,2,2.52} {2.18,2,2,2,2,2} {2.36,2.18,2.18,2.25,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_sup_flat y1 y2 y3 y4 y5 y6) (--dih3_y y1 y2 y3 y4 y5 y6 + #0.0076 > -- #0.956317 + #0.419124 * (-- &2 + y1) + -- #0.0753922 * (-- &2 + y2) + -- #0.252307 * (-- &2 + y3) + #0.5 * (-- #2.82843 + y4) + -- #0.246082 * (-- &2 + y5) + -- #0.788717 * (-- &2 + y6))`; idv = "2763799127"; doc = "date{2010, 8, 4, 13, 47, 22.850389}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum with input mDihedral3 {2,2,2,sqrt8,2,2} {2.52,2.52,2.52,3.0,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_sup_flat y1 y2 y3 y4 y5 y6) (dih2_y y1 y2 y3 y4 y5 y6 + #0.0047 > #0.936544 + -- #0.636113 * (-- #2.1 + y1) + #0.140759 * (-- #2.1 + y2) + -- #0.0771734 * (-- #2.3 + y3) + -- #1.29068 * (-- #2.82843 + y4) + #0.591328 * (-- #2.1 + y5) + -- #0.0521775 * (-- #2.1 + y6))`; idv = "5943578801"; doc = "date{2010, 8, 4, 13, 50, 57.176690}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input Dihedral2 {2.1,2.1,2.3,sqrt8,2.1,2.1} {2,2,2,sqrt8,2,2} {2.52,2.52,2.52,3.0,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; let apex_std3_lhh = define_dart `apex_std3_lhh y1 y2 y3 y4 y5 y6 = [ (&2,y1,#2.18); (#2.18,y2,#2.52); (#2.18,y3,#2.52); (&2,y4,#2.52); (&2,y5,#2.52); (&2,y6,#2.52) ]` ;; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_std3_lhh y1 y2 y3 y4 y5 y6) (dih_y y1 y2 y3 y4 y5 y6 + #0.0012 > #1.01332 + #0.148615 * (-- &2 + y1) + -- #0.379006 * (-- #2.18 + y2) + -- #0.379441 * (-- #2.18 + y3) + #0.583676 * (-- &2 + y4) + -- #0.184708 * (-- #2.25 + y5) + -- #0.18471 * (-- #2.25 + y6))`; idv = "1836408787"; doc = "date{2010, 8, 4, 18, 41, 52.689213}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input Dihedral {2,2.18,2.18,2,2.25,2.25} {2,2.18,2.18,2,2,2} {2.18,2.52,2.52,2.52,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_std3_lhh y1 y2 y3 y4 y5 y6) (--dih2_y y1 y2 y3 y4 y5 y6 + #0.0059 > -- #1.33909 + #0.0724529 * (-- &2 + y1) + -- #0.486824 * (-- #2.18 + y2) + #0.317329 * (-- #2.18 + y3) + -- #0.00479451 * (-- &2 + y4) + -- #0.751179 * (-- #2.25 + y5) + #0.350857 * (-- #2.25 + y6))`; idv = "1248932983"; doc = "date{2010, 8, 4, 18, 42, 54.778495}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input mDihedral2 {2,2.18,2.18,2,2.25,2.25} {2,2.18,2.18,2,2,2} {2.18,2.52,2.52,2.52,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; (* Aug 5, 2010 *) add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_flat y1 y2 y3 y4 y5 y6) (dih3_y y1 y2 y3 y4 y5 y6 + #0.0046 > #0.88473 + -- #0.443946 * (-- #2.36 + y1) + -- #0.244711 * (-- #2.1 + y2) + #0.205592 * (-- #2.1 + y3) + -- #0.739126 * (-- #2.55 + y4) + -- #0.127198 * (-- #2.1 + y5) + #0.61582 * (-- &2 + y6))`; idv = "6725783616"; doc = "date{2010, 8, 5, 6, 6, 10.752609}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input Dihedral3 {2.36,2.1,2.1,2.55,2.1,2.0} {2,2,2,2.52,2,2} {2.52,2.52,2.52,sqrt8,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_std3_hll y1 y2 y3 y4 y5 y6) (--dih3_y y1 y2 y3 y4 y5 y6 + #0.0116 > -- #1.30119 + #0.392915 * (-- #2.36 + y1) + #0.142563 * (-- #2.1 + y2) + -- #0.258747 * (-- #2.1 + y3) + #0.417088 * (-- #2.45 + y4) + -- #0.0606764 * (-- &2 + y5) + -- #0.637966 * (-- #2.45 + y6))`; idv = "9185711902"; doc = "date{2010, 8, 5, 6, 25, 0.685185}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input mDihedral3 {2.36,2.1,2.1,2.45,2.,2.45} {2.18,2,2,2,2,2} {2.52,2.18,2.18,2.52,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_std3_hll y1 y2 y3 y4 y5 y6) (dih2_y y1 y2 y3 y4 y5 y6 + #0.0011 > #0.957661 + -- #0.250506 * (-- #2.36 + y1) + #0.145114 * (-- #2.1 + y2) + -- #0.0549376 * (-- #2.1 + y3) + -- #0.0384445 * (-- #2.45 + y4) + #0.5275 * (-- &2 + y5) + #0.118819 * (-- #2.45 + y6))`; idv = "6284721194"; doc = "date{2010, 8, 5, 6, 29, 20.593944}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input Dihedral2 {2.36,2.1,2.1,2.45,2.,2.45} {2.18,2,2,2,2,2} {2.52,2.18,2.18,2.52,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; add{ineq = `!y1 y2 y3 y4 y5 y6. ineq (apex_flat_h y1 y2 y3 y4 y5 y6) (dih2_y y1 y2 y3 y4 y5 y6 + #0.0042 > #0.868174 + -- #0.701906 * (-- #2.25 + y1) + #0.136514 * (-- &2 + y2) + -- #0.209239 * (-- #2.18 + y3) + -- #0.493373 * (-- #2.65 + y4) + #0.537385 * (-- &2 + y5) + #0.0187672 * (-- #2.2 + y6))`; idv = "3137600529"; doc = "date{2010, 8, 5, 10, 13, 11.823742}\n ineq generated by Mathematica holineq\n Generated by generate_ineq_datum_p with input Dihedral2 {2.25,2,2.18,2.65,2.,2.2} {2.18,2,2,2.52,2,2} {2.52,2.52,2.52,sqrt8,2.52,2.52}.\n The inequality has been fitted to cfsqp margins."; tags = [Cfsqp; Xconvert;Lp; Lpsymmetry]};; end;;